I was trying for a diplomatic win, but five turns before I would have had the UN built, I hit the cultural win. Most unexpected. All I did was randomly set up cities where they look good, assimilate every city I can, never, EVER raze a city, and beat everyone else to every wonder. Each and every wonder. No matter how worthless, build them all.
I'm still a bit miffed about that UN thing, though....
Yeah, I'm almost tempted not to choose that option as a victory condition. On Huge maps it's too easy for me to get a cultural victory (100,000 cp) . I try to build all the wonders, on the harder levels you have to give up on the early ones, then start building the wonders when you know your ahead in tech. I don't randomly put my cities anywhere, but I hate deciding where to put them so they aren't too close, and not too far away. So I set up a pattern that I place my cities, regardless of resources, etc. I place all cities three squares away on the diagonal. This leaves just one expty square in between four cities and there is on average of four squares per city that are given up to another city when it's up to it's 21 square potential. Easily enough to support 16-20+citizens. The only thing that screws up my pattern is water squares and mountains

. But this would probably only work on Pangea maps, because Island maps you would want to make sure there were more cities on the coast.
I don't think trying to achieve a cultural victory is a good idea. However, you still need cultural structures. The first thing I build in my cities (except for the defender unit) is the temple. This increases the borders and makes citizens happy - an important thing if I don't have access to luxuries yet. If the city gets at least 10 commerce points I build a library and a university as soon as possible. Cathedral is the last cultural structure I build. If your city is rich, you can also build a market (it gives commerce and happiness), instead of colloseum (too expensive to maintain a colloseum in all cities).
I build all the improvements in all cities (by the time you get all the way to building a collesuem, your city should be near size 12, and should be making enough money to easily support it, especially if you are in Republic or Democracy). My usual production order: (aquadects, courthouses varies by city, so I won't put those in)
1. Temple
2. Library
3. Marketplace (for the commerce and happiness bonuses)
4. University
5. Bank
6. Colleseum
7. Cathedral (If I have the Sistine Chapel, then I'll move this up the list)
I suppose you could put number 4 ahead of 3 if your not worried about cash reserves, same with number 5, it could go lower on the list.
1) My total cultural point is now 10800 on the domestic adviser screen. But how do I find out my current cultural point/turn (overall and in each city)?
On the cultural advisor screen (F5), it tells you your total of all cities at the top, then it will also list each city, just select the city, and it will say when each improvement was built, and how many culture points (cp) it has added to your civ total. You need 20,000 in one city or 100,000 in your entire civ to win.
2) In ancient era, should I concentrate on building wonders or on buliding city's infrastructure (temple, library, etc)?
If a cultural building has been completed for more than 1000 years the culture from it doubles (Temples would produce 4 cp instead of 2 every turn), so if you can build these quickly it helps because it doesn't take long before those 1000 years are up in the early game. I focus on expanding first, you want as many cities as you can before the AI claims all the land, then immediatley build those cultural improvements. More cities = more cp /turn (as long as they can support the improvements)
What I do on the Huge maps is have my capital build something like this to start (depending on situation):warrior, warrior, settler, warrior, settler, temple, wonders. The settlers it built will be your next settler/warrior factory until your done with your expansion, and hopefully that temple will keep your citizens happy until you can get some luxuries connected to it. This works great if your capital is built on a river, or if not, if a nearby city is I'll have that one build the first wonders (free aqueduct, so hopefully you'll reach 12 citizens fast, and have enough production to build the wonders quickly). With this stategy, you shouldn't have a problem on Chieftain building all the wonders in your capital (except the coastal ones, unless you are on the coast). Having most, if not all the wonders, plus all the cultural city improvements in all your cities, and the AI doesn't stand a chance. So he'll get mad at you and try to squash your culture crusade with his military, so make sure you are well defended.