Jesusin's article above is the best. Here are some quick pointers, assuming trying for early culture dates:
1) You can win with just 3 cities, but 9 cities are ideal. The reason is to build the religious wonders in all 3 cities. In order to do that you need 9 cities to all build temples for each religion you have.
2) For Legendary cities, I generally go with a mix of 1 or 2 super cottage cities and a GP farm. The GP farm usually lags behind but you will use Great Artists great works to bump the culture.
3) Cottages are ultimately king. Try to roll starts with good Bureau type cap locations...lots of riverside green or flood plains and enough food and hammers to grow and build culture buildings. You simply can't beat the massive amount of culture per turn you will get in each cottaged city once you switch to Free Speech and have the culture multipliers in place.
4) Beeline Music for religious wonders and Sistine
5) Try to keep your high Food GP farm pristine as far as Great Artist GPP. Good thing is National Epic gives great artist points. You can build Great Artist wonders here as well or in your cap. At some point you will run Caste and Pac to pump Great Artists out the wazoo.
6) Often your Capital is the best cottaged city. Make sure all religious wonders are built there asap as well as Hermitage. This city should hit L fairly quickly on its own merit. This allows you to dump Great Artists in the slower cities to keep pace. Try to balance the Great Artist missions so that all cities hit L within a few turns off each other. If City 1 and 2 are hitting L around the same time but City 3 his hitting L 20 turns or more later then you are doing something wrong. The victory screen is an easy way to keep track of the progress of each L city. (Important: Note the difference between Total Culture in each city vs. Turns to Legendary. Just because City 2 has more culture currently than City 3 doesn't mean it will hit L faster. This is important for determining which cities to hit with Great Artists to balance completion dates)
7) The 3 to 6 cities other than your 3 L cities are basically what I call "helper cities". That is, there only purpose is to help you win culture so you can abuse them. First and foremost, you want to settle the helper cities in such a way that they overlap the cottaged L cities so they can help work cottages for the L cities while those cities grow or whip in culture buildings. This ensures that your cottages mature nicely by the time you hit Liberalism and run Free Speech. It's not farfetched to get between 500 and 1000 cpt in the cottaged L cities if you do this. (Don't overexpand/conquer. 9 cities is usually more than sufficient if you place them correctly. However, 1 or 2 more cites may be ok to make sure cottages are constantly worked or to grab that marble off in the wilderness. Too many cities can hurt your economy in the timeframe of a culture game and prevent you from keeping your slider at 100%)
Next, the helper cities should have plenty of food on tiles that aren't overlapped to cottage tiles. The reason: so they can a) whip stuff like temples and missionaries like crazy b) later you will starve out Great Artists in as many cities as possible. The GP farm will pump a lot, but there is no reason not to be able to get more GAs in helper cities. Really 1 or 2 of these helper cities should whip out temples and mishes like mad. You wont have to worry about stacked unhappiness later when you go culture mode.
8) Great People: You might pump out a first Great Scientists or two from your cap or other city for an academy to boost your way to Lib or a Philo bulb. Otherwise, you want Great Artists exclusively.
9) Tech path: Obviously the usual stuff early - worker techs/strat techs. Then bee-line Music, although Great Library is not really necessary. Then simply head straight for Liberalism and take Nationalism for The Taj (golden age boost for more GPPs and it gives great artist points, plus golden age boost commerce which boosts culture) and The Hermitage. Using Oracle for Code of Laws and Conf is always a good play.
When you hit Lib, you should be at a point where most culture mulitpliers are in. Switch to Free Speech and Caste, if you are not already in it. Pacifism if you are actually running a religion (usually one that spread to you for diplo reasons) Merc is not bad if you have banking for more artists and, ofc, Rep if you built the Mids, but you would already be in Rep.
At this point you are ready to basically turn-off research and run your culture slider as high as possible. 100% culture is ideal, but sometimes you have money issues. Build wealth in as many cities as possible to boost the culture slider. Trade techs and/or beg for gold from the AIs and sell resources for gold. With culture slider high you won't need any happy resources, so trade them for gold or health, if you need it.
Your cottaged L cities should have grown to about max now and are working all mature cottaged tiles. Helper cities are building wealth or starving out Great Artists.
The GP farm L city can also starve out great artist, it's getting a lot of culture from Sistine artists anyway.
10) Wars: You generally want to avoid war altogether, so diplo is very important. If I switch to a religion (which is good for OR and Pac), I wait for them to spread from the AIs and then spread them around. You usually want at least 3 different religions in your cities - the more the merrier. I usually only found Conf and/or Pac, possibly Theo depending on an early GP pop from Oracle.
However, you may want to war early to grab an AI capital or two, which often make good L cities. Probably a good idea to kill these AIs off since it can come back to haunt you later when you are weak. Only keep other AI cities if they can contribute nicely as helper cities (overlap, food, key resources.
You may get weaker later once you turn off tech, so you want to ally up to a group of AIs and possible start fake wars to keep villains off your back. Only do so if necessary though. You want to have some defense, but you want to get away with as little military as possible so you have more gold for the culture slider.
11) Marble is the obvious resource of choice for culture. So many helpful wonders for culture use it. Stone less so and Mids is not vital. In fact, I would only consider Mids if I had stone and a way to get it up quickly. Early Rep is always a boost, but you get engineer points. A GE could be used on Sistine or Taj though. Also, Stone and Copper boost some of the religious wonders.
12) Lastly, as mentioned, Lib/Nat is generally where you turn tech off and run culture. However, if you are still in the process of getting in all the culture multipliers, there are other techs that are useful. First, Printing Press is really something you should get asap as it boosts the cottages. Constitution is nice if you aren't in Rep. Biology is great for the food boost. Otherwise, once you flip the switch it's should be a quick race to hit Legendary in all 3 cities.
Later game culture wins would focus on things like Sushi and Eiffel tower.