Draco is the son of a rich, pureblood wizard family. He's also spoiled, prejudiced, and envious. He's in Slytherin House, and he's Harry's arch-nemesis at school. In the first book, Draco's snobbery takes center stage. He alienates Harry right off the bat by saying that Ron isn't the right sort to associate with. Yet in Book 2, it's Draco's anti-Muggle-born prejudice that becomes more important. After all, Draco actually calls Hermione a Mudblood, a nasty name for magical people born in non-magical families. This insult is bad enough that Ron tries to hex Draco to spit up slugs (sadly, his broken wand sends the curse back to Ron). Draco's first response to seeing Petrified Mrs. Norris and the Heir of Slytherin message written on the wall of one of the Hogwarts corridors? "Enemies of the Heir, beware! You'll be next, Mudbloods!" (8.173). So Draco makes his views known pretty clearly.
Of course, given Draco's outspoken hatred of Muggle-borns, it's not surprising that Harry and Ron suspect Draco of being the Heir of Slytherin. Using the Polyjuice Potion to disguise themselves as Draco's two best friends, Crabbe and Goyle, they are disappointed to learn that Draco isn't the Heir. The thing is, Draco talks a good game about death to Mudbloods. Yet he doesn't do anything actually violent in Book 2. He brags a lot, he's a bully, and he's a coward – but, as of Chamber of Secrets, he's all bark and no bite. Lucius Malfoy seems genuinely dangerous to us, but Draco is still just a mean little kid. What he'll grow into, we don't know yet.
We do get to see more of the influences on Draco in Book 2, as we get a sense of what makes him the insecure, nasty jerk that he is. His father is as cold as a block of ice, and Draco is obviously both desperately eager to, and incapable of, pleasing him. Draco is hugely jealous of Harry. He takes every opportunity to taunt him about his fame, about Colin Creevey's photos, and about Ginny Weasley's unfortunate poem. Still, Draco's jealousy of Harry makes him obsessed; he becomes so focused on laughing at Harry for his weird flying during the Gryffindor/Slytherin Quidditch match that he misses the Golden Snitch "hovering inches above Malfoy's left ear" (10.79). So Draco's loathing for Harry is actually distracting him from accomplishing the things he wants to do.
It seems to us that at least part of Draco's problem is that he can't figure out who he is outside of the Malfoy name. When his father succeeds in removing Professor Dumbledore as Headmaster of Hogwarts, Draco struts around as though it's his success. When Professor Dumbledore returns and Lucius gets fired as school governor, Draco looks "resentful and sulky" (18.131). All that seems to matter for Draco is the Malfoy name and Malfoy views; do we ever see him trying to distinguish himself from his family?