From Civilization V Wiki
To win a cultural victory, you must acquire five complete "branches" of social policies – that is, you must own all policies within five different branches. Once you've done that, the "Utopia Project" is unlocked. Construct that project and you win a cultural victory. You cannot rush or purchase this project. It must be constructed.
For every additional city you control, it costs 30% more culture to open your next policy - this encourages you to have fewer larger cities.
- The French Civilization gains a bonus to culture until a certain technology level is met in the mid game
- The Aztec Civilization gain culture for killing units
- The Indian Civilization has happiness benefits for having fewer larger cities
Cultural Victory Objective: Completely fill out five different Policy Trees.
You're not likely to win a cultural victory before the Technology-level that disables the French Nation's Culture bonus.
Another likely candidate is the Indian nation, which receives a hefty happiness bonus by way of a passive counter for population density unhappiness. However this comes with a penalty for unhappiness caused by multiple cities.
Therefore, a good way to achieve this victory type is to go the Indian Civilization approach, and to stick with a small number of cities (3 seems to be a good number, as suggested by the achievement that involves winning a game with this nation and victory type).
Note that the above strategy is perhaps crucial, because you do indeed receive a scaling penalty for each new Policy for each new city that you create. That means that a small number of overly developed cities is a better strategy than a large number of cities producing small amounts of culture towards a growing Policy cap.
Each city should be located fairly near one another, but a larger spacing that normal (Around 10 hexes) is good for claiming a respectable chunk of land for yourself, and for keeping other Civilization out of your borders.
Each city needs to be withing a large area of viable resources. Coastal tiles are probably not the best choice unless there is a good amount of water resources available, or unless it's an inlet that allows you access to a large number of land tiles anyway.
Each city should have a healthy amount of both Production rich squares, and food rich squares. Luxury resources are extremely valuable and important, and should be developed near to if at all possible, as they help produce extra happiness (Which can be upgraded within the final tiers of the Commerce Policy Tree), and is probably the leading factor contributing in a Cultural Victory.
-Food: The more people in your country, the more potential for happiness. With its bonus, the Indian Civilzation will receive more happiness and therefore more culture per citizen (Assuming other factors aren't making your people unhappy) than other cultures. Therefore, each of your three cities should try to become as large as possible, making food on as high of a priority as any other resource. More citizens also = more research = more buildings that increase culture output.
-Production/Hammers: Research and the building of culture % modifying buildings are all dependent on production units, so it is vital that you settle somewhere with a good number of viable production hexes. Mountainous areas, especially those rich with Gold, Silver, Gems, etc. are very good options. Usually a town will have more trouble producing production that they will food so plan accordingly.
-Units: At the start of the game, pump out 3-5 workers and set them to shaping the land around you to a food/production rich environment. If your towns are growing at the population/culture rate that they should be, they won't quickly run out of things to do, as more and more hexes will become workable, and there will be more citizens to work them. Military units are virtually pointless in a Cultural Victory. If you navigate the political scene in a friendly way, you should be able to avoid war, and therefore the need for military units. Try to keep your starting warrior alive, and that might possibly be enough to keep barbarians (Assuming you're playing with them turned on) away from your workers. Your towns can fend for themselves should the barbarians come close. Also keep in mind that all units cost upcome, and that will slow your research and growth progress. Tread carefully.
-Buildings: I'm actually going to suggest that you put production/science buildings before cultural buildings. If you have selected good locations for your three towns, and are developing them correctly, you should never have to wait more than a building or two to get to the cultural structures. Just remember that the more production % modifying buildings you have, the faster you'll be able to produce cultural buildings. One hand washes the other. Same goes for science buildings, with more of those, research goes faster. More research + more technologies that open up new cultural buildings.
-Research: Just like Civ IV you can't just go down one path, you'll eventually have to backtrack and go into other areas to progress farther down the more culturally-centered part of the tree. Start with pottery and mining, then go for writing and make your way towards education. Only pick up things from the bottom side of the tree that allow you to progress farther down the non-military technology side of the tree, though you won't be able to go all the way until post-Dynamite. Basically anything that helps culture, science and especially production - you want.
-Wonders: Fortunately most computers (And humans, for that matter) don't really go for culture Wonders. They don't have a lot of use otherwise, and anyone going for any other victory route will basically just grab policies as they come, rather than aiming directly for them. This definitely plays to your advantage. Grab Culture Wonders first, but also race for free techs, free Great Scientists, Free Production, etc. Leave Great Pyramids, as the Great Library is a much better early option and it will become rather useless later anyway.
Politics and Surviving:
City-States: My advice on this is to leave them alone. Even with all the +Rep bonuses you'll glean from your policies, in order to see a significant culture pay-off for befriending city-states, you'll be spending a truckload of gold that could otherwise be spent on far more valuable things. They are never aggressive anyway, unless you provoke them by trespassing, etc. So it's best just to be casual acquaintances with these kids.
Other Nations: Nations aren't as aggressive as they were in Civilization IV, but they will still take advantage of weakness if they detect it. A good way to overcome this is to simply never open your borders to them. They react a lot less negatively to this than they have in the past, and it will keep them from knowing how militarily weak you are. This is aided by your out of control border growth boosted a la culture production. Feel free to enter into Research Agreements with Nations if you have the money, but never agree to go to war with them (Where are you going to get the military anyway?), and I would even warn against declaring friendship or defensive pacts, as you're likely to stir other hornet's nest by doing that. Since you will also not be having military units to station on their borders, they really _Shouldn't_ ever have a reason to attack you, unless you give it to them - don't
Tech Advantage: Despite the fact that you'll be purposefully using a small number of cities, you'll quickly out-tech every other civilization. This is usually another good deterrent against aggression, and is, in fact, a real defense against unwarranted attacks, as you should be able to quickly whip out much more powerful units against their horde of pikemen/musketmen.