One thing to keep in mind is that the AI in CiV is highly competitive. Even the most peaceful seeming of allies will confront you if they think you are nearing victory. In the end, it takes a lot to keep an ally honest and good, and some... you just don't stand a chance. Civs like Russia are nearly guaranteed to be hostile no matter what when it's crunch time.
Catherine, Alex, Monty, and a few of the other warmongers are obvious choices for civs to watch out for, but a couple of other ones that normally wouldn't fall into the obvious alarm category stand out to me. The ones I've had bad experiences with are Hiawatha, William Van Orange, and Wu Zetian. All of them have their way of being problems. William reacts VERY poorly to being beaten to wonders or competing for city states. I've had him denounce me out of the blue despite a lengthy peaceful coexistence, supported by common religion and trade. Wu Zetian is incredibly opportunistic. If she gets a significant military advantage over you, or if she can dogpile you with a few of your neighbors, it hasn't mattered if she was friendly. Hiawatha... he's unique in that he's actually very easy to keep friendly or at least neutral, but he will snowball very easily if left unchecked. His expansion flavor is very high, and in the end-game, wide is highly advantageous.
A couple of others that I really watch out for are the religion-spammers. Gajah Mada, Ramkamhaeng, and Gandhi, if they get a religion, will flood you with stupid amounts of missionaries. You'd think there's no way for them to crank out 5 great prophets and about a dozen missionaries in around 50 turns or so, but somehow, it works. Boudicca combines that tendency with a high aggressiveness as well.
Ultimately, there are very few other civilizations which you can pretty much ignore. Gandhi, if he by some freak of chance doesn't get his own religion, is one of those. Extremely easy to keep friendly, very low expansion flavor, low competitiveness even for wonders (which he is willing to spam.) Almost any civilization you run across you have to treat as a potential enemy.
So in answer to your question... the identity of the AI's I run across influences my planning and strategy a lot. Then again, the ways of dealing with most of the AIs can be quite similar. If there's someone with three great prophets invading my pristine faithful cities, I don't care if it's India, Indonesia, or Spain, I'm capturing them and leading a war of attrition to get a good peace deal. Free holy sites. If Alexander, Atilla, or Genghis are right next to me when I start, I'll focus on building a few more ranged units than I usually would to defend, and possibly prioritize Iron or Horses more than I usually would in settling new cities. There are not many civilizations that for me warrant a special strategy just for them. The Zulus are one because of the outstanding nature of their UU and their utter willingness to use it. They are very, very dangerous, and so I will address them specifically.
