/sigh
Where is Deau when we need him?
The important thing to remember about CV's is that it is not about trickling out tourism, it is about late game concert tours. If you are winning at T400, then you are trying to overwhelm the AI through raw tourism, which only will work on low difficulty.
So here is the more or less "optimal" method;
3 city growth/science focused tradition start. You need an extremely tall capitol, and you need respectable beakers (700 -1100 is fine). You are going to need a lot of policies, so 3 cities is often better than 4, but you should build 4 if possible. Keep in mind that you need more hammers for CV than any other victory, so do not settle more than 1 low hammer city - you need to get culture buildings up asap to overcome the policy penalty for the extra city.
Play more or less like a typical science victory. Bee line to scientific theory - then right to Radio (bulb with Oxford) and take Freedom (Freedom is usually best for 3 city CV). Then bee line to Plastics, and right up to Internet (you can detour to biology before plastics, but usually better to get it after plastics), then back track to Radar (yes, Radar last).
Your build order is the same as for an SV, except you want to get the Hermitage, the Writers guild, and the Artists guild up as quickly as possible. Do not build the Musicians guild til very late, and do not waste hammers on wonders that use musicians. On Immortal/Deity, the AI will out-spam your archeologists with no trouble, park a unit on all digs outside your territory until you can go dig them up - you can steal one dig per enemy with no diplomatic hit. On high level, you will have trouble filling 4 cities + wonders from digs, but you should be able to fill 3 cities with artifacts.
On Immortal, you can reliably enter renaissance with printing press, build Pisa, then the Sistine, then Uffizi. On Deity, you will have to decide based on who is on the map, but it is sometimes worth faith buying a GE, entering renaissance with acoustics and bulbing Sistine.
Policies: Full tradition, aesthetics opener, Exploration opener (for Uffizi), Rationalism/secularism, Freedom to tier 2 (New Deal).
The rest is map dependent. If there are plenty of cultural CS's and you have a lot of cpt, you will want to finish rationalism next and bulb the finisher on Plastics or Internet. Otherwise finish aesthetics, and then take the tier 3 Freedom tenet Media Culture - this should time out nicely with getting your towers up.
Use your GW's for bulbing policies, though making great works with the 1st two is fine. Collect GA's and make great works in batches to get theming bonus, you will spend most of your time in Modern, so you should be able to easily get same era theming easily.
It takes ~18 turns to make your first musician after you build the musicians guild if you do not staff it. Time the finish of the musicians guild so it finishes ~19 turns before get Radar and rush buy airports. This timing is critical, you do not want any musicians before you have airports.
If the cultural leader is not next door, settle a new city close by and gift it to them. Now just generate and faith buy as many musicians as possible and send them in for concert tours. You might need to DoW if they will not grant open borders.
General tips:
Never, ever, under any circumstances, give open borders to the AI (never!)
always buy open borders from all civs
send a TR to every civ even if you get less gold for it
send your spys as a diplomats into civs with differing ideologies.
The Inca (by far the strongest civ for hammers/growth) are probably the best for CV, Poland and the Shoshone are also extremely good, and the Aztecs are very good choice (growth).
Do not neglect faith. On high difficulty it is better to plant your prophets and let the AI give you religion.
On Deity and sometimes Immortal, there will be a runaway culture civ. You will have to just go conquer them. You should never have this problem on Emperor or below.