I think there are two types of threats that the AI can pose: they can outright kill you or they can runaway with score/culture/science/etc. Military AI are primarily only very threatening if they are right on top of you, but sometimes they kill everyone and eventually get to you. In addition, there are AIs that are just
<snip> and kind of just constantly make your life a living hell.
Military AIs are usually worst at the beginning. civdata.com lists all the AI traits, the AIs with the highest values (8 or 7) are really problematic and can make higher difficulties unwinnable. Early game Attila on Deity with a good start is honestly unbeatable sometimes. He comes at you T50 with 8 horse archers and 3 rams, nothing you could have done. Shaka can be pretty horrible too. Fortunately, some of the military AIs are really incompetent, particularly Assyria, but others kind of suck too. Genghis Khan supposedly has high bias but he's never been a problem for me personally. Montezuma is another problematic one, Dido tends to backstab in my experience, and Poland deserves an honorable mention for being very dangerous - with plains start and very high likelihood to train military they will have a good army if they attack.
For runaway AIs, pretty much any civ is capable, but there tends to be a few that are usually more problematic than others. The AI usually does better with huge empires with a ton of cities, so civs with higher expansion rates do better. Shaka is the most expansive AI by far, so he can do really well although I haven't had severe problems with him. One of the most problematic is Hiawatha, he expands a ton, and for some reason his expands work. Randomly, I've had a lot of problems with Suleimann - not a particularly good civ but he expands a lot and it seems to work for him. England does well in most games I play, not sure way. Babylon and Korea, by nature of their high affinity for science and their ridiculous UAs can be runaways. If you aren't close enough to kill them early you might have real problems with their science. I've had the same problem with China, they don't have the natural science boost but some of their strongest traits are growth and science, which leads to very tall and hard to take down empires.
AIs get very annoying when they expand aggressively, are obnoxious with city states, and generally passive aggressive with everything that happens. AI with extreme warmonger penalty are usually bad, but if you're playing peacefully they will do away with other warmongers and be your friend. Ramkhamhaeng is horrible because he hates warmongers, and will call you out any time that you do anything with a city state. Alexander is pretty similar, except he is more expansive, but hates warmongers less, and will generally be a bit less of a dick. Maria Theresa is really annoying marrying and allying city states. Backstabbing is really annoying, and for some reason I've found Carthage doesn't mind backstabbing. Been backstabbed by Denmark a few times too.
The best AI to start near are non expansive, do not build units, but build wonders. Ramesses is the best, Enrico Dandolo, Gandhi, Pacal are all very good too.
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