A few tips when doing the cultural victory condition:
1) It's generally much easier to play smaller maps and fewer civilizations to achieve this victory than any playing larger maps and larger number of civilizations. The reason for this is more relating to the fact that you need to establish easy trade routes with other civilizations and having trade routes with other civilizations is a necessity for for the Tourism boost.
2) For early game advantage, try to rush Great Library and Stonehenge. You not only get the early game advantage of the big Faith boost, but you also obtain your religion earlier than other civilizations and once you obtain your religion, spread at very aggressively to other civilizations. You'll get a large tourism boost for having the majority religion and more so if you make it a World Religion during the World Congress. The Great Library gives a nice scientific boost as well that will allow to you reach several eras much faster than other civilizations and quickly adopt an Ideology you want quicker. Any Ideology works to obtain Cultural victory, but I personally find Freedom to be the best choice as they have more perks for the cultural victory than any other Ideology.
3) You do NOT want to build a lot of cities to do cultural victory. Although it may seem a bit tempting to build a lot of cities, it's actually very detrimental on trying to build the necessary wonders where they grant you Great Artist/Musician/Writer points with the respective Great Work of Art/Music/Writing slot. Although it's possible to obtain cultural victory with a lot of cities, it's gonna have a lot detrimental effect to your empire's happiness rating.
4) For AI civilizations, choose the ones that are easy to befriend. Having the AI's "likeness to be friendly" 6 or higher makes it easy for you to obtain this. Polynesia, Dutch, Shoshone, America, Venice, India, Iroquois, and Siam are relatively easy to befriend civilizations. Likewise, having AI that has a good habit of being a warmonger (like Mongols, France, Japan, Aztecs, and Huns) or having a serious backstabbing disorder (*cough, cough* Chinese) makes it REALLY difficult to achieve cultural victory.
5) Speaking about the World Congress/United Nations and proposals, there are a lot of resolutions you can pass that can make obtaining cultural victory much easier. Arts Funding, World's Fair, and International Games are the top three proposals you should consider proposing and a lot of civilizations are very happy to vote for these resolutions. ESPECIALLY Arts Funding because from most of my playthroughs, I've saw a lot of civilizations that are pleased with me proposing Arts Funding while all of them get pissed off for me proposing Science Funding (I think you get diplomatic penalties for proposing things that will get them angry or go against their interests IIRC and can make obtaining cultural or diplomatic victories harder).
6) For social policies Adopt (in order) Piety, Tradition, and Aesthetics.
Personally, I didn't find cultural victory to be THAT hard, you just need good adjustments in your map settings and a bit of luck. In fact, I manage to obtain a cultural victory as India by only having three cities (thus obtaining the Bollywood achievement). I personally find trying to obtain Domination and Diplomatic victory to be harder (particularly the later if you are playing Venice AND trying to obtain the One City to Rule Them All achievement at the same time).