Civsassin
Immortal
Again, it may not be a cultural playstyle but any game that could wind up a dominion victory could also be a culture victory. Kill everyone except leave one capital. Wait. Build culture victory thingie. It's no harder than dominion, but will obviously take a little longer.
Or you could play the Songhai with a small base of real cities and go conquering. Puppets love to build that UB and the culture will come rolling in. You can easily have more than three real cities because your puppets will produce so much culture you'll make up the difference. Any civ can do this, but the Songhai are the best. Their UA means they have lots of gold to rush more military, so the beatings are easy and, like I said, the puppets all will go right for that awesome UB. The advantage here is you don't have to kill everyone. Just try to grab a bunch of puppets. ~20 should do the job in a jiffy.
First off, you cannot raze capitals, so you can't just kill everyone and build the culture thingie. You have to trade the capitals to someone to get rid of them or you have to keep them if everyone else is dead.
Moreover, fully exploring five different policy branches takes a long time. If your spending your time building units and unit upgrade buildings, you probably aren't building cultural buildings. You have to plan early for a culture victory. Each additional city to your capital raises your cultural requirement by - I believe - 33%. To win a cultural victory, you also need to build policy enhancing buildings - Oracle, Sidney Opera House, etc. You will have a very difficult time funding nearly constant wars and focusing on culture.