There's two main Byzantine cultural victory paths:
One is the sacred sites gambit. With it being largely specific to Byzantine and much more difficult to apply to other civs, I'll mention that first.
This is Piety first. You'll want a faith pantheon. Ideally, you found a religion first, but being early will suffice.
When you found the religion select one with religious buildings as a follower and in addition select as your bonus belief an extra follower one that also provides religious buildings.
Next you want to enhance the religion as soon as possible, when you do select yet another religious buildings follower belief. (For a total of 3). With all faith for a long time going to be tied up with religious buildings, select Religious Texts as the enhancer so your religion will spread to your own cities without needing any Missionaries.
You need to be first to reformation, select Sacred Sites. This will give you 3 full sets of buildings with tourism bonuses.
Upon finishing Piety, you'll get a free Great Prophet but will already have enhanced your religion, settle that one for culture, faith, and gold which will speed up religious building construction and culture.
The second tree with then be Ascetics. In this case, finish the full tree as soon as you can to both increase tourism bonuses for open borders / trade routes and double theming bonuses.
Third is to open Exploration just to allow construction of the Louve.
In this case, Order is likely to be your best ideology.
The theming world wonders are less important than standard culture for this approach but would still help. By contrast, early world wonders providing faith my be more useful than normal.
Standard Hotel/Airport/National Visitor Center/Internet apply
Two: The standard cultural victory path, but with a religion
Pick whichever of Tradition or Liberty you are more comfortable with as the first tree and complete it.
Use standard tactics to found a religion early. (This can be any of Faith pantheon, faith natural wonder, Stonehenge, religious city states among others.)
If you've gone Tradition, then you can faith buy Eiffel Tower extremely early.
Upon founding a religion, decide if you'd be best off with a second pantheon, second enhancer, or third follower.
It folds directly into standard cultural victory for everybody from here. (Aesthetics , theming bonueses, hotels, airports, national vistor center, internet)
Things that people sometimes forget to do that will help any cultural victory:
1. Play the exchange great works with AI mini game, which will make it much easier to get the theming bonuses. As a couple of examples:
Oxford requires 2 Great Works of Writing from different eras and different civs not the city owner. If you didn't exchange, you'd have to conquer 2 different AIs capitals to get this.
Hermitage requires 3 Great Works of Art from different civs and eras, but your own civ can be one of them. So again, if you didn't exchange you'd have to conquer 2 different AIs capitals to get this.
2. Swap great works around to get those theming bonuses, and also note that if one of your cities has a hotel and the others don't, that great works should be moved to the hotel city. (same applies to airport present and especially national visitor center)
3. If you've finished the aesthetics tree, you can faith buy Great Musicians in end game. Doing this while hosting the Olympics is a really good idea. The other good time is after the Internet is discovered and National Visitor Center is built. And after you reach industrial era you can save great points up. (Say 2500 points and faith buy 2 Great Musicians the same turn)
4. Normally the best thing for the first World Council vote to be for someone seeking a culture victory is Arts funding to speed up Great Works. You want Worlds Fair in the second World Council to allow more time to get production up in your core cities. Olympics is usually desirable as soon as it can be proposed.
