World background: Prepare for huge recap! Alternia is an alien planet, a harsh wilderness world looked over by two moons and a viciously bright sun. It is an ugly planet, a troll planet, and it is out to kill you. It is inhabited by what appear to be a number of sentient species, but by far the most numerous and the ones in charge are the trolls. Or their young, anyway; adult trolls are not allowed on Alternia, making it a planet of (deadly, uncontrolled) children. Due to the sunlight being very unpleasant for most trolls (and also partly due to the hordes of undead that appear during daylight), most trolls are nocturnal, sleeping during the day in slime-filled vats known as recuperacoons. The night is roamed by musclebeasts, who are also deadly. In fact, most of the flora and fauna of Alternia is probably dangerous, or prey, or both. Trolls are the top of the food chain, technically speaking. As you can guess, they are also pretty damn nasty, including to each other. Most of the adult trolls are off-planet, fighting in a great intergalactic war, but the young trolls are still pretty deadly.
Trolls are vaguely humanoid in appearance, having the usual complement of limbs and sensory organs, but they have grey skin, yellow sclera to their eyes, and horns that are coloured like candy corn. They have an insectile life-cycle, starting as eggs and hatching into grubs, then pupating into somewhat more humanoid forms. They also have brightly coloured blood that (usually) ranges from maroon to purple; this is known as the hemospectrum and a troll's place on it determines their importance and social class. Unfortunately for Karkat, he isn't actually on the scale. He's a mutant; his bright red blood puts him below even the lowest of official blood-colours. As such, he is slated for immediate culling should anyone ever discover his blood colour. At the top of the scale are the purple-blooded trolls, who are sea-dwellers and have some small difference in anatomy; at the very top of those is the Empress, Her Royal Condescension, who is one of only two trolls to have fuschia blood.
Trolls technically start life as a bucket of genetic material, collected from adult trolls in one of the two sex-based relationships (more on that later). This is taken to the mother grubs, who reside deep under the planet's surface in the brooding caverns. The mother grubs combine thousands of said buckets in one giant genetic slushie, and then lay the eggs that hatch into larva. After wriggling around for a while, troll larva spin cocoons on the stalactites in the caverns, and pupate. After this, life only gets more difficult. Firstly, a newly hatched troll must go through a series of dangerous trials before they even get to leave the caverns; then they must be chosen by a caretaker, or lusus, of another species before they are allowed to the surface. After which, they travel to a new location, whereupon the troll may take up residence in a hive. Some trolls get to design their own, some don't; Karkat was one of the trolls who did. After this, the major points of a young troll's life is to find and settle into relationships with other young trolls, before being (probably) sent off-planet to take part in the giant interstellar was of conquest all the adult trolls are, in whatever capacity, involved in to some degree.
At this point, a divergence into troll romance is unfortunately necessary: trolls believe in 4 different possible romances, known as quadrants; these are divided into relationships based on "pity" and "hatred" along one axis, and reproductive and non-reproductive along the other axis. A relationship based on reproductive urges and pity, which is about as close as trolls seem to get to a regular human romance, is known as matespritship and is represented usually by a red heart symbol and known as a redrom or flushed relationship. A relationship based on hatred and reproductive urges is a kismesissitude, usually represented by a black spade; it's usually called a blackrom or caliginous relationship. A relationship based on hatred but non-reproductive is an auspistice, which works slightly differently from other relationships in that an auspistice is actually an intermediary between two other trolls, preventing them from flaring up into full-blown kismesis. It is represented by a black club symbol and known as an ashen relationship. Finally a relationship that is non-reproductive but based on pity is a moirallegiance, in which a dangerous troll finds themself attracted to a troll of a calmer temperament who works to balance the excesses of the more dangerous troll. Note that calmer does not really mean less dangerous; in fact, a troll who is too calm may be unable to restrain their moirail, which makes for a bad moirallegiance. Moirallegiance is known as the pale quadrant, and represented by a red or pink diamond symbol. An adult troll must be in both of the reproductive relationships, lest a collector drone come calling with his bucket. The punishment for being unable to fill a bucket to the drone's satisfaction is instantaneous culling; thus for trolls romance is literally a matter of life and death. Then there's quadrant flipping, where a relationship is so unstable it in a state of constant flux between two different quadrants, most often red/pitch. Canonically Karkat has had both a moirallegiance with Gamzee, and a matespritship/kismesis with Terezi. He's also had a one-sided hatecrush on John, and may have been in some kind of weird ashen relationship with himself and Jade.
Asides over, when Karkat was 7 sweeps old (approximately 15 1/2 years old)
Personality: Much like certain ogres, trolls have layers. Unfortunately, in Karkat's case, every single one of those layers (or at least the visible ones) is full of rage, irritation, and pretty much every other synonym for angry a thesaurus can provide. Karkat is highly opinionated, intolerant of perceived stupidity in others, impatient, and prone to extended and highly metaphorical insults and empty logic, which fall like a rain of cinders on anyone unfortunate enough to be near the metaphorical volcano that is Karkat. He comes across as arrogant, unfriendly, and generally unpleasant to know.Karkat is prone to taking offense to the smallest thing, and will do a triple axel into a swandive off the metaphorical handle on occasion, in which case the best thing to do is step back and leave him to it.)
The further into the layers you go, however, the more that umbrage turns inwards. He hates himself more than nearly anyone else that might be on the recieving end of the verbal abuse stick; partly because of his blood color, partly because of his own perception of himself as a failure. He notably hates his own past and future selves to the point where he feels like he needs an auspistice to mediate. Unfortunately, because he is afraid of screwing up, he tries to hard and expects too much of himself, leading to, yes, frequent failures (or perceived failures). His blood color makes him a mutant, and his tendancy towards failure make him insecure and feed into his paranoia.
This leads neatly into the next point of Karkat's personality; he is ferociously paranoid. Most trolls are at least a little paranoid, one would suspect, living as they do on a planet where death is cheaper than a discount store chocolate bar, but Karkat takes it to whole new levels. His blood colour comes into this, of course. Whether it's true or another product of Karkat's paranoia, he firmly believes that his mutant blood color, if ever revealed, will cause his to be instantly culled. He is also paranoid about being seen as anything less than completely competent at anything.
From this it can be pretty clearly seen that Karkat is pretty much constantly seeking approval and/or validation. This is why he constantly seeks to be the leader and to be better than anyone else. He hates to fail, but is constantly setting himself up for it; in turn this feeds his paranoia, misery and anger. He has no respect for herself, instead trying to gain it from those around him and being pretty transparently desperate about it. He tries a little too hard, pushes a little too far, and it all tends to collapse around his ears. Zero self-approval ratings all round, really.
Buried somewhere deep underneath all of that is somewhat less of a jerk, who is rather deeply protective of the people he cares for even if he can't see it, and who is, ever so slightly a hopeless romantic who dreams of serendipity and romance. He will freely hand out romance advice to people, for example, even though he is not good at taking advice or analysing his own relationships. And even if he does care for someone, he will still insult them; he just means the insults less.