To do this you must simply build the United Nations wonder in one of your cities to end the game. This becomes available after you have accumulated 20 Great People, Wonders or converted cities.
Things that improve your culture level most are having large cities (more people), building temples and cathedrals. Building walls can also help to stop your city being converted by another civilisation. And finally what may seem obvious, keep hold of any Great People you come across (even use spies to steal some from other civilisations) and build Wonders.