Ok I'm pretty sure you can win a cultural victory with any number of cities, but there are 2 good reasons to stay small:
1.) If you stay small, each of your cities is already developed. This means you get to build other things and don't have to waste time setting up cities to pump out more culture than they cost. You also don't have to find happiness sources to make up for the base city unhappiness
2.) Many of the social policies that boost your culture are more effective for small empires.
But if you build every cultural building and fill every artist slot, I'm sure you could produce more than enough culture to make up for the increasing costs. You would just be wasting a lot of time you should probably be using to build a defense, science buildings, and growing your population.
As for conquering opponents, in the sense that each city you capture only helps your culture (if you puppet, not annex), then yeah that's a good idea. However, conquering opponents also means you have to spend time building a military and finding happiness sources to cover your puppets, so I think it is still optimal to play peacefully and tall.
Ok, now for some specific recommendations.
There are times when you shouldn't play cultural. These are when you have a civ that has no bonuses towards that style or your starting territory is not good enough to support 3-4 very good cities. If you both have a civ that is good for going tall/culture AND your starting territory is good, then you should consider a cultural victory.
At this point, think about which 5 policy branches you would want to choose. You always want Tradition, Piety, and Freedom. If you do not choose these, you may still win, but probably not culturally... So other options are Liberty, Honor, Patronage, and Commerce.
Even if you pick Liberty and/or Honor, I would recommend filling out Tradition ASAP. And most if not all of Piety immediately after. By then you should unlock Freedom, and again you should fill that out. Note this is only a recommendation, I think there are a few extremely unique situations that would suggest mixing and matching. Most of these involve holding out on Legalism (the 4 free culture building policy) until you have cities up and running with base culture buildings. As the Aztecs, for example, Honor will help you utilize an early defensive army, and your aqueducts from the Tradition finisher (the main reason to finish Tradition ASAP) may in fact be overkill when you have Floating Gardens. Now you can hold off on Legalism until you set up 3-4 cities and get Drama for Amphitheaters.
Always fill out every artist slot you can, and use Great Artists for Landmarks. Try not to fall behind in tech or military, though obviously you can't have it all. I usually find Libraries and the National College to be enough science to stay just behind the pack, and several early Archers or Composite Bowmen and a few Spearmen can be enough to hold off armies.