For Domination:
Japan. Bushido owns the battlefield. When you hit the late medieval period, start churning out Samurai (without overly neglecting buildings) and take your continent.
China. The Great General bonus is nice, but the real reason I'd go with China is the Cho-ku-no. A crossbow with logistics (2 attacks per turn) out of the gates is devastating.
Germany, Ottomans, Persia, Huns, Mongols, Rome, etc are all good choices too.
For Science:
The Mayans. Free great people are nice, but the real bonus here is the pyramid (I'm assuming you have G+K). +2 science early on is a huge bonus.
I'm sure some will disagree with me here, but I think that Austria is a nice pick. The coffee house helps you churn out Great Scientists quickly, and Diplomatic Marriage lets you pick up some nice high population cities. When you annex a city-state, you get all their units, too, so it's a nice way to ward off AI DoWs and pressure those whom you want to take over (annex a city-state on the other side of a civ while at war, after they've moved their units away, proceed to cause chaos).
To be honest, most civs can pull out a science victory pretty well. I've heard many vouch for Babylon and Korea, but I've never played as them so I can't necessarily say one way or the other. I really liked Sweden as well, for reasons I'll get to later.
For Diplomacy:
Sweden. I thought they'd be a pretty awful civ, but their UA is actually quite useful. Playing Sweden successfully requires some skill, though. You have to set up a moderately wide, very tall empire and focus on churning out great people late game (they are a VERY late game-oriented civ). Fully explore Patronage, and use those great people to ally the world. The only thing I dislike about this is that it will really irritate the AI, so be prepared for war, not just on your own soil but on your city states (you have to protect them from being conquered to get their votes). Keep an eye out for Mongolia... The Carolean is really nice, as well. March is always good.
Greece. I haven't personally played as them, but they are always a supreme pain in the rear mid to late game due to their UA. 100+ influence in 2/3 of the world's city-states? Come on...
For Culture:
Egypt. In my opinion, Egypt is the best for a cultural victory. You will get most of the wonders, if you play your cards right. Focus on a tall civ, with about 5 core cities. Conquer if you want (I'd take at least 5 more cities), but puppet everything. You don't want a higher social policy cost. Take Tradition, Piety, Patronage, Liberty/Commerce (I prefer the latter), Freedom. Place your cities in positions to be easily defendable, as you will be a target with all those wonders. Cities next to mountains will help you to keep up, or even get ahead in science, but don't sacrifice a fertile area for a mountain. With only 5 cities, you need to make each one count.
I've gotten my best victory with Egypt, winning a cultural victory by 1890 or so.
Ethiopia. Their UA will help you stave off land/wonder-hungry, aggressive AIs.
Siam. Naruesan's Elephants are tanks. Play the city-states; they will help you grow, since you are going to play to a tall civ. Patronage is a requisite here.
Again, most civs can manage a cultural victory moderately well.
That's my two cents. I feel like I'm forgetting some civs, but what's there is pretty good, I think. These are the civs I prefer; others my have different strategies, which may work better or worse than mine. Civ V is very much a personal game; as is often said, there's more than one way to skin a cat.