Karkat is a crab in more ways than one. He constantly insults others in the style of a stereotypical internet troll, supporting his arguments with what seems to be empty logic. Jade has even gone to the point of flipping out at Karkat.
In keeping with the crab apple asshole theme of his personality, he rarely smiles (one of the few times he does can be seen when he deals with Gamzee). He is also often seen screaming, pounding his fists against his keyboard, or generally raging when talking to the other trolls. He is not particularly patient as when explaining himself, he makes even his most sound advice appear to be random insults and he becomes visibly aggravated when other people are having fun.
Somewhat similar to Jack Noir, Karkat is ruthlessly ambitious and will grasp at any opportunity for power, but he just doesn't have the subtlety of his troll compatriots and ends with the sharp end of the stick more than once. His real strength is to shepherd his boneheaded troll friends into following a plan, which just gives him another excuse to go off the handle when something does not go according to it.
Despite his flippant demeanor, he also seems to care quite deeply for his friends' well being, judging from his tearful reaction to Sollux's apparent death, his seemingly genuine desire to help Eridan and Tavros with their relationship issues with Feferi and Vriska, respectively, and his willing enough to revive Kanaya.
Towards the end, Karkat seems to show a softer side that hasn't really been shown before. In his final memo, future Karkat leaves a rather emotional story shortly before Kanaya responds to it. At this point he treats her in a friendly and polite manner, something that he really hasn't done with anyone before.
Karkat is not nearly as bloodthirsty as some of the other trolls (like Vriska or even Kanaya), and has trouble jumping to action when shit hits the whirling device. If things go off plan and he's not nearby, he gets pissed off and starts yelling at the related parties. If he -is- nearby he faints, starts crying or sits there looking stupid.