Many of the leaders' personalities seem to be tied to their traits.
For example, leaders with the Spiritual trait will give you bigger diplomatic demerits for running a different state religion than their own; non-Spiritual leaders usually don't care about this as much. Conversely, Spiritual leaders will give you more diplomatic bonus points, and more quickly, for sharing their religion.
Aggressive leaders are usually more dangerous in the early game. They tend to respect other leaders with that trait more, and will spam units. By the mid to late game, though, the unit spam and emphasis on researching military traits has cost them, and they're usually far behind in techs and easier to deal with. (Although he's not Aggressive, Sitting Bull also falls into this category, but he's usually easier to get along with than the Aggressive leaders.) Ragnar is the exception to this; as he's Financial as well as Aggressive, he's usually able to keep up in the tech race. It's usually best to take down Ragnar early, or to befriend him if you can't.
Protective leaders seem to like to "turtle" a lot. It's not that they won't get involved in wars, just that they're better at defense than offense and tend to play the game that way as a result.
The Industrious leaders have an advantage in going after wonders, and tend to like building them. With Industrious leaders I find I not only have to avoid tech trading something to them that enables access to a wonder, as I do with all civs; I also have to be careful about trading that wonder's prerequisite techs. Industrious leaders often seem to bee-line to a wonder tech when they get close to it.
The Philosophical leaders are generally, I find, the easiest to get along with in the game. The exception to this, however, is unquestionably Alexander; his Aggressive trait trumps his Philosophical one in this regard. He'll turn on you as soon as he has the slightest advantage.
Many of the leaders have also earned reputations as "back-stabbers"--that is, a predilection for declaring war on you even if they're "Pleased" or even "Friendly". Some of the worst in this regard seem to be Catherine, Joao, Mehmed, Alexander, Huayna, and Qin.
Certain leaders are "psychos"--they'll declare war even if they're at a significant disadvantage (usually in terms of technology rather than overall power). Montezuma, Isabella, Tokugawa, and Shaka seem to be the worst in this regard. Shaka and Montezuma, I find, tend to be the most dangerous neighbours to have in the early game--they'll build up a large force of units and declare war on you very early on.
Conversely, a few of the leaders are extremely easy to get along with and very forgiving of past transgressions. Gandhi, Hatshepsut, and Mansa are at the top of this list.
Mansa is also a fiendish tech trader and can be useful in that regard, however, he's also adept at research, so if you trade too many techs to him you may find yourself falling behind him in technology. Gandhi is no slouch in that department either. Huayna, Wang Kon, and Ragnar are also formidable techers.
