I was actually going to start a similar thread yesterday, but got busy at work.
I'm considering the following: some civilizations just click well with the Deity AI bonuses or have UAs which don't require intelligent use.
Really strong:
Greece (can spam classical units to dominate early. Has lots of gold by virtue of Deity AI which it turns into CS spam. Alex as a leader is the most competitive thus he will always exploit weakness if he so perceives it. He can probably crush any other Ai civ neighboring him assuming an equal start)
Siam (spams a religion and tons of gold, which it turns into doubled CS spam. Can snowball if left unchecked)
Russia (the AI boost to happiness, gold and hammers means she can actually make use of the Krepost because the AI tends to not prioritize border growth. So Cathy ends up with a huge empire and plenty of hammers and resources for an army ifs she wants one)
Babylon (the most powerful civ in the game coupled with Deity AI bonuses rarely fails and usually is the tech leader, first to industrialize and quite frequently a very good warmonger with superior units)
Incas (the terrace farms give the Ai enough hammer that it can just wondermonger like mad. Also, Pacuiti is fairly warrior like, so he can stomp some face if he wants to or needs. His lands make it hard for early warmongers to cripple him)
Austria (The Ai can have 12000 in the bank. Meaning it can swoop over and take CS whenver it feels like. The bonus to GP generation really helps it since it usually makes horrid city spot choices.)
Rome (the Ai bonuses make it strong like Russia, meaning it goes wide as hell. Also it can warmonger early with its two UUs pretty effectively. But it has bonuses to road building and satellite city infrastructure, which means that its wide empire will be more efficient than that of a standard bloated Deity AI empire)
Spain (AI isabella makes great use of the free extra settlers and scouts. If she ends up near good NWs she can snowball so damn hard it's not even funny)
Just plain annoying;
Monty, Shakka, Attila nad Genghis - the Deity Ai bonus makes them able to amass a horde and cripple or wipe out a few other civs. They rarely transition into a power mid to late game but boy can they be a pain early on!
Ghandi because Ghandi
The Celts because they are ALWAYS snagging the good tenets.
Arabia/Morocco because they steal DF and Petra from you and make you break your keyboard with your face.