UPDATE: Although you may still find my old strategy guide helpful, I would like to refer readers also to jesusin’ more current and comprehensive post on cultural victories,
Cultural Victory on the Higher Levels.
I would also like to add that based on current experience, I no longer recommend building the Great Library at all (too many great scientists instead of artists), and I only recommend the Pyramids in certain situations (like when you are actually pretty sure you can build them). With regard to the Pyramids, when in doubt, don’t build them. The Sistine Chapel and the Parthenon are probably the most important wonders, though no wonders are necessary. I would also like to add that for the fastest win, I recommend quiting at Liberalism and using it on Nationalism. You can ususally pick up Printing Press from spare beakers after switching to all culture. Note also that while this strategy recommends buying many cathedrals and temples, the fastest results at the highest level now seldom do this and just build them instead.
I would like to thank jesusin for all he has done to elighten CivFanatics about Cultural victories in the GOTM threads, in his own post linked above, and in the comments in my post below.
On to my old guide, and I hope you find it useful!
—–
At the request of some of you fine Civfanatics, I’ve finally gotten off my lazy butt to do a write-up on how I won my six HOF Deity games (three on Beta, three on the new one). Be warned–it’s still quite a challenge with this strategy, but it works fairly often for me, proving that contrary to the conventional wisdom, it’s quite possible to achieve a totally peaceful builder’s win on the highest settings. In fact, it’s only really challenging on Deity. So this one goes out to all you builders out there: peace wins. 
Settings
I’m going to explain how to do this using strategic settings, but it DOES work with organic settings–it’s just harder, and it may be impossible with some types of maps and/or sets of opponents. You do need one thing for this to work, especially with chopping having been nerfed: you need stone in your starting city to build the Pyramids, and you must have the Pyramids to win before the AI launches the ship. I have done this without starting stone before the patch, but have found it impossible with 1.61. So if you don’t find stone in your starting position, regenerate the map until you do.
Disclaimer: this strategy borrows bit and pieces from many others’ strategies and tips. I am especially indebted to the excellent players, who contributed to the conversations in the Beta Gauntlet III.
For a peaceful win, I’m convinced that Elizabeth is the best cultural victor, hands down. The combination of financial (for cottage spamming) and philosophical (for the great artists) can’t be beat. The next best cultural leader is Mansa (half priced temples plus financial). Also, the slower game speed the better for faster wins.
EDIT: Huyana also excels at this strategy, but with some modifications (not all peace). For more, see the conversation in the thread below.
Maps: smaller is easier but larger offers the opportunity for faster wins. Why? Because the more opponents you have, the harder it is to keep the peace and to build the Pyramids on Deity. But more opponents also means more tech trades and so a faster win.
The easiest map allowed by the HOF is Pangaea on smaller maps, and Pang ea or Continents on Standard and larger. The easiest map, period, is Balanced, because it offers great resources, and it is the largest map, but the new HOF doesn’t allow this map type. Water means slower contact with the AI, which means no early tech trades. Yes, it’s easier to expand with a continent all to yourself, but it’s nearly impossible to tech fast enough to win without trades.
Opponents: You want peace. This is easier with Peacemonger opponents, but it can be done with more warlike opposition (though with Izzy in the game, it may be impossible). This strategy guide assumes that you already know how to use diplomacy to keep the peace. That’s a whole topic in itself and deserving of its own thread. If you really want to win though, here are the peacemongers, in approximate order of peace-lovingness: Mansa, Cyrus, Hatshepsut, Asoka, Gandhi, Catherine, Frederick, Louis. Also, this strategy assumes you know how to cottage spam.
More settings: low sea level helps A LOT to expand to the requisite six cities. If using Pang ea, a "pressed" coastline, as I understand it, presents "sub-continents" (which = good! cut off your opponents!) but without a lot of islands (which require naval investment = a waste); but I personally enjoy "random" — call me sentimental. (Temperate or Tropical both work fine.)
There are three main obstacles to a peaceful cultural win on Deity: building the Pyramids before the AI, expanding fast enough to get six cities, and achieving legendary culture before the AI launches the ship. Seldom will the AI win by any other means using this strategy.
First City Build Order
This all depends on a successful start. It’s pretty easy after that. The challenge is to both build the Pyramids and not get boxed in. It’s not easy, but this seems to work best: Worker > expand to population size two while building a warrior (you can finish him later) > as soon as size two, switch to Settler > expand to size three while finishing the warrior/starting new one > Worker #2 > Pyramids. After that, I build whatever seems most necessary until the "temple spam" stage.
I never build any barracks or any troops other than one warrior per city. I don’t hook up with copper until I’ve built six warriors, so that the option is present to build warriors and not the more expensive axemen or spearmen. If diplomacy fails and someone invades (rarely happens), retire, rinse, and repeat.
Starting Research Path
Bronze > Masonry > Wheel > Agriculture > Pottery > Writing > Alphabet
Teching after that is a matter of researching techs that the AI is slow to attack, so that you can use them to trade. So, for example, I never research Calendar or Mathematics, because the AI always researches them, and I can trade for them, whereas if I research them, I can’t trade anything for them because the AI already has them. So techs like Paper for example that aren’t a big deal in most games become important to research relatively early, because the AI researches them late, and you can trade them for others. You need to make early contact with all or most of the AI to establish tech trades so you can win.
What to Do
First of all, be aware that in a peaceful world on Deity, the AI will launch their rocket any time between 1500 and 1850 (or even earlier perhaps on larger maps). The more opponents you have, the faster the AI launches the ship. On a standard map with six opponents, expect 1500-1700. The AI finishes faster on larger maps. You will need to finish early to win peacefully, and sometimes, the AI just outraces you. If you can manipulate them into war with each other diplomatically, that’s a good thing.
At first, micromanage your capital with an emphasis on food and growth if not building a worker or settler. Don’t worry about defense. Even if you play with barbarians turned on, the land gets settled so fast that the fog of war vaporizes, along with the barbs.
Use your first worker to chop two or three forests to accelerate Settler spawning. As soon as you finish Masonry, quit chopping and improve the stone square. In addition to being required for the Pyramids, this helps a ton with speeding up settler production. You don’t need roads until just before switching to Pyramid building, and if you’re lucky and both the stone and your first city are on a river, you don’t need roads at all until much later.
By now you’ve finished the first settler. Settle in a spot that will allow you to cottage spam later and which blocks off and claims potential territory from your nearest opponent the most effectively. Build another settler right away. Move worker number one with settler number two, as soon as the worker finishes its task when settler two is spawned. Improve one food square if in cultural boundaries and then chop until the second worker is about done in the first city. Time your first worker’s arrival back to your capital with the production of worker number two. Hook up to stone using both workers. Switch to Pyramids the second the stone is hooked up and don’t stop until you finish it. If the AI beats you to the Pyramids, rinse and repeat. After the stone is hooked up, have one worker mine all of your hills while the other chops everything in sight. You want to be as large as happiness allows to occupy those mines. You may need to improve a food resource to have enough food to support your miners. Put everything you’ve got into finishing the Pyramids, because if you don’t, it’s over.
The reason you need the Pyramids isn’t so much for the happiness boost from Representation (though it’s nice)–it’s so you can buy religious buildings later, without having to tech all the way to universal suffrage.
Meanwhile, city number two needs to produce settler(s) ASAP. You need six cities to win, but only three of them need to be good cities. A "good" city in this strategy is one that you can cottage spam with and which is connected to fresh water. Health resources are more important than luxury resources. All of the general resources are useless except stone, marble, and copper. Stone you absolutely must have to build the Pyramids. Stone, marble, and copper are the resources that double production speed of the major religious buildings like Cathedrals, Academies, and Mandirs, which is why they are nice to have. Only stone is essential, though. After building the Pyramids and establishing three cities that you can cottage spam with, you’re over the hump.
It’s a challenge on Deity even to get three decent and three marginal cities without war, but it can be done. Really once you get that third one, you’re usually home free, because you can often squeeze three more cities in around the edges of your cultural borders somewhere. Don’t worry if your opponent’s borders encroach upon your marginal cities. Believe, me you will beat their borders back big time in the end and before they flip you. In fact, if an opponent builds close to your capital or one of your three culture cities, you can usually flip it by the end game, which sometimes has provided me with my sixth requisite city.
Try to cut off your opponents from access to a peninsula, even if there are no resources on it. You can use it later to place those marginal cities. And don’t worry if your three other cities occupy crummy positions. In the end, it really doesn’t matter that much. Only the cultural cities matter in the end.
You will almost always receive at least two engineers from the Pyramids if you don’t dilute the engineer gene pool. Use the first one on the Great Library, and use the second one on either the Sistine Chapel or the Taj Mahal. Don’t build the Great Library in the city you want to use to spawn Great Artists later.
In a perfect world, you will manage your great people production so that you receive exactly three scientists from the Great Library before your Great Artist farm overwhelms it. Use all of your scientists to build academies in your three cultural cities. The cultural bonus they provide is just as valuable as the science boost. An academy in each cultural city provides double the value of being a civ with the cultural trait.
After building the Pyramids, focus on cottage spamming, occupying as many cottaged tiles as possible continually while growing. Don’t worry about amassing culture in your beginning game. Almost all of it will come in your end game, when you turn your culture bar up all the way.
The overall plan is: cottage spam and tech as fast as possible; shut off all technology at a certain point (more on this later) and keep the culture bar as low as possible without having unhappy citizens, maximizing your gold input; meanwhile buy religious buildings; then crank up the culture bar. I know this part of the game plan is familiar to many of you, and I certainly don’t claim originality here.
After other civs switch to Emancipation, you will probably have to turn up the culture a little to maintain happiness. Remember, the idea is to increase commerce and BUY all the religious buildings, so it’s fine to not have much culture until the end. After you quit researching, the AI will gift you some techs, often good ones. That’s the bonus of being a "backwards people." 
I struggle with the question of when to quit teching. It seems to me the options are, in successive order: right after liberalism (for Free Speech), after Nationalism (for the Hermitage), or after Printing Press (for the bonus gold). My best game was when quitting after Printing Press, but I got some lucky gift techs in that game. Still, I’m leaning toward thinking that is best.
If possible, one of your cities that is not producing culture needs irrigated farms and food resources to be your Great Artist factory (you usually need caste system too). Any artists produced up until about 400-500 AD I add to one of my cultural cities (usually the slowest one). After that, save them for a big round of culture bombs at the end. The culture bombs are great at helping overcome discrepancies between culture levels in your three cultural cities. Everyone who is serious about their cultural games will be interested in O.H. Dog’s superb Civ IV Culture Calculator Spreadsheet. It really helps in planning out your end game.
I never try to found early religions, and I am seldom able to found the late ones. But getting them to spread is usually not a problem in the end. You don’t need all the religions, but the more the better. The fewest number I have won with is three religions. Whatever you do, NEVER use Theocracy. You MUST let those religions in! It’s much easier for religions to spread on land maps. Trade routes also seem to help.
If there is one overriding world religion, obviously I switch to it, but if there is even one nearby country of an alternate religion, I remain without. I switch to Pacifism only when: a) my neighbors are so pleased with me that the "heathen religion" factor doesn’t matter; or b) late in the game when the AI starts switching en masse to Free Religion. Remember: only cities with your state religion receive the great people bonus under Pacifism.
I buy or build every possible major religious building in my three cultural cities. For those who don’t know, the six city requirement results from the temple building factor. On small and tiny maps, six temples in any one religion allows you to build three major religious buildings (cathedrals, etc.), and each major religious building provides a 50% bonus to your cultural production. On standard maps, six temples allows you two build two major religious buildings, and if there is a change in this on the larger maps, I’m not aware of it.
Switch to all culture, no income as you approach the end of building your major religious buildings. You don’t want to waste your end game production in your cities, so it’s OK to finish the last major building in each city with hammers, as opposed to buying it. If you want to estimate what your cities’ cultural rates will be when you switch into all-culture mode, turn the culture slider up all the way for a second and check them out on the city screen. Don’t forget to slide it back down after you’ve checked. This will help you to plan ahead. Or for a more precise and sophisticated look: use the culture counter spreadsheet I mentioned earlier.
The major religious buildings are what wins the game. So let’s imagine that you receive a modest four religions with six cities on a small map. That means you can buy or build FOUR major religious buildings in EACH cultural city, providing a total plus 200% culture bonus in each of them. One will usually get the Hermitage, too. Combine this with cottage spamming, the other culture producing buildings, any added great artists, and perhaps a wonder; and you can easily create cities that produce 600-800 points of culture per turn by the end game. It doesn’t take long to win at that rate.
EDIT: That was a conservative estimate on culture rate. My last Deity game, I had a city amassing culture at 1,080 / turn as I approached victory.
There are many other nuances, but this post is getting long-winded already, so I’ll leave it at that for now. Maybe we can use this thread to elaborate on the strategy and improve it. I would love to hear your questions, advice, criticism, etc.
Good luck peacemongers! Someday, we take over the world. 