I find that it actually varies somewhat from time to time, how the computer acts, but that is probably due to the fact that every game is bound to progress differently than the last one. As for Civ's I myself like or dislike, well...
The Russians, Mongols, Spanish, Zulus & Japanese tend to always end up as my mortal enemies whenever they are present. They always have huge armies and bite after more than they can swallow, generally my whole empire...
Civ's I tend to get along *better* with ( that is to say only OCCASIONAL meaningless wars isntead of permanent ones ) are the Babylonians, Chinese, Americans, Egyptians & Romans. Frankly, that's the case.
Though I either tend to random the enemy AI's or choose Civ's who historically go together, such as picking the "English, Spanish, French & Russian" or "Greek, Persian, Egyptian, Babylonian & Roman" and so on...