Build a temple as soon as you have the technology for it. Starting with the technology helps, and building that
first, even before warriors will give you a massive head-start to a cultural victory. That is assuming that your cultural victory is planned to be achieved in your capital.
Obviously, prioritize techs that lead to cultural buildings (like libraries, unis, colloseums, etc) Also, if you build the buildings sooner, then they will gain you much more culture in that city compared to waiting for those buildings to be built. Therefore, be prepared to rush-build as many buildings as you can in that city as soon as humanly possible. If in despotism, sacrafice your population for the glory of your empire. If you then dont have enough pop in that city, get surrounding cities to make a worker or two and add them into your cultural city. Some workers on "Prepare to be sacraficed" duty near your cultural city, especially when a breakthrough is near, is also a good idea.
If in Monarchy, Republic or Demo, save up gold to ensure you can rush-buy any buildings as soon as the advance is achieved.
Finally, defend that city to the death. You will basically be focising most of your game on getting that single city with as high a culture as you possibly can. If that city is captured, then your chance of a cultural victory is severly diminished. If they raze your city, them all hope is lost. So ensure you keep that city.
