I understand that "most annoying" is not the same as "most effective", but I'd still have to say that Alexander's AI is both annoying and effective. Now I wonder whether it would be possible to make an AI that is effective without being annoying, since the annoyance might come simply from the fact that the AI is strong and may be winning. I don't think that's entirely the case, though. For example, they could remove Alex's tendency to call you up every 10 turns just to insult you without his AI becoming any less effective.
Alex allies all the city-states and, as a result, has large cities, advanced technology, plenty of social policies, lots of happiness, lots of faith, and a large military all at the same time. He often overpowers other AIs in my games, completely destroying a few (one time, his entire continent). Not strong enough to defeat me, a human player, of course, but he makes it harder. And once I wipe him out everyone else still alive hates me, even though I just did them all a favor.
But you have to remember that ALL AIs in the game basically fit the criteria for clinical sociopathy pretty well. They're almost always only out for themselves (even if they don't even do it well sometimes) and they're utter hypocrites. It's true that some are more peaceful and/or friendly than others, but even the most pacifist ones will declare war if they think you're weak enough and they covet your land enough.
Proximity is another factor. As others have already pointed out, even the most peaceful will become more hostile if you're near them, while warlike civs that are far away can become your good friends. Simply having adjacent territory can for all intents and purposes be considered a negative diplomatic modifier.
As mentioned, some AIs are more peaceful than others, but that doesn't mean they can't be annoying. Peaceful and friendly AIs are annoying in different ways - spamming their religion (Ethiopia), spamming wonders (India), putting cities in the weirdest places (Polynesia), being technologically advanced and a legitimate threat to the space victory (Korea), and so on.
Oh, and Elizabeth is practically impossible to befriend no matter what you do, but at least even when she dislikes you she doesn't call you just to insult you like Alex does.