Teddy: One of the easiest AIs in the game to get along with. Will likely be friends with everyone except his immediate neighbors.
Saladin: A natural enemy to anyone who's likely to found a religion, especially Philip, but also Gandhi, Pericles, Gorgo, Hojo, and Peter. Qin too, if he manages to get Stonehenge. Always friendly with Mvemba though.
Pedro: Hates anyone who generates more great people than he does, but not many rivals can. Montezuma, Victoria, and Mvemba seem like the most likely candidates for earning his ire.
Qin: Certain to hate Cleopatra and Catherine every game.
Cleopatra: She likes anyone who's aggressive, so almost everybody. Will always hate Gandhi though.
Victoria: By far the most difficult AI to get along with. Likely to be enemies with everyone, including you, every single game, unless you get lucky enough to have her right next door.
Catherine: Since the AI doesn't seem to prioritize counterspying much, which is the only thing she hates, she's likely to have lots of friends.
Frederick: Hates Pericles and probably Gilgamesh too.
Pericles: Hates Fredrick and probably anybody who starts nearby him.
Gorgo: Gandhi's eternal nemesis.
Gandhi: The AIs are always declaring war on each other for stupid reasons, so he's likely to hate everybody except you. Unless you're playing aggressively of course. Then he'll hate you too.
Hojo: Since most of the civs have unique infrastructures that generate culture or faith or both, he doesn't seem like he'll be too difficult to get along with. Likely to end up having the same preferences as Cleopatra.
Mvemba: The opposite of Saladin, basically. Expect Philip and Saladin to fight over him constantly.
Harald: Will always be spurning Cleopatra's unrequited love. He likes naval civs, so Victoria, Pedro, Fredrick, and Philip.
Trajan: The reverse of Pedro and Qin, he only respects civs who can beat him at his own game, which is almost nobody. Might get along with Peter every once in a while.
Peter: Loves Qin and Saladin. Otherwise, he's a total crapshoot.
Tomyris: Hates everyone for the same reasons Gandhi does. Might make friends with Gilgamesh if they start close to each other.
Philip: Surprisingly easy to make friends with. Will otherwise have the same preferences as Saladin.
Gilgamesh: Super friendly by design. Not much else to say.
Montezuma: Easy enough for a human player to get on his good side by just selling him luxuries, but the AI probably isn't smart enough to figure this out, so he'll hate everyone.