Troll4Hire, I've been using the Aztecs a lot before Mongolia came around. It is true their Jaguars are exceptional but the only problem is that every civ begins to hate you from the start. Don't know why but I've always had 3 or 4 civs declare war on me within the first 70 turns.
According to my experiences, no matter how "friendly" you playstyle is, sometimes AI players decide that you're in their way. And when they do that, they tend to bring some friends as well.
I played a game recently with England where I tried to stay passive as long as possible, and only attacked my neighbor (Sweden), when another AI player (USA) asked me to join his assault on Sweden. I did, but the States tricked me there: Washington chickened out, and denounced me a few turns later for going to war vs Sweden. After capturing several Swedish cities I was up against the US, Babylon and a few remaining Swedish cities. But what I'm trying to point out that I got a big red "Backstabbed You!" text next to both Washington and Nebuchadnezzar. So I just wanted to stress out the AI itself can be kind of a jerk sometimes too, not just humans.
Not sure if this can help, but here are some general ideas as well, for example:
In the early stages when you're running circles with jaguars, you probably encounter a few City-states as well.
- If you're going to war versus several city-states your influence-resting point will drop at several other CS-s too, because you're pushing around the weak and defenseless. I assume AI players you've met will not tolerate this behavior either in general.
- If you're attacking just one specific CS, but it has a protector civ (an AI player pledged protection), that specific player will probably hate you for doing it, and he might tell the others on the map that you were messing with his CS-s. So keep an eye on which CS is protected by whom when you're demanding tributes.
- If you're in a multiplayer game with both AI and human players on the same map (we do that from time to time), you have to accept the fact that you cannot trust anyone. The AI players will throw away hundreds of years of cooperation, declaration of friendship, even active trades for around a 500-1000g cash and/or resources. (but the fact that you cannot trust these guys goes both ways - he will declare war that instant, but sometimes he just sends 1 or 2 unit to patrol your borders, and that's it. He took the other player's cash, and waging war against you... on paper.

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You can get a few hints why the is AI hating your guts (contested borders, he wanted to expand but you put cities there, you build Wonders he wanted, he got informed that you had spies in his cities, etc). This is all there in the diplomacy panel, check it from time to time - you won't find answers for every question or event happening on the map, but it's enough so you could start guessing.
Sry if this was too "generalized", but I don't really know anything about your experiences with civ5.

These are some really basic examples of playing with AI players, but other than that i find it challenging to give tips, since i don't have a clue what is happening under 70 turns in your games.
