What are Hamlet's most notable character traits? Do you see these traits as generally positive or negative?

Hamlet's character has been an enigma to readers and theatergoers for four hundred years. Hostile critics such as Leo Tolstoy—the great Russian novelist disliked Shakespeare's plays overall—and T.S. Eliot noted numerous inconsistencies in Hamlet's behavior which, in their opinion, made the entire tragedy defective as a work of art. In the age of film we have seen wildly different interpretations of Hamlet by actors ranging from Laurence Olivier to Derek Jacobi to Kenneth Branagh (and others). There is no straightforward or definitive answer to the question of whether Hamlet is a "realistic" character, or whether his traits are largely positive or negative. The puzzle about him is real, and there is no obvious solution.
That said, we can still make some observations about Hamlet and argue that both the character and the play are so iconic and enduring that the criticisms of Eliot, and others, are irrelevant. Most readers and audiences respond to Hamlet, in my view, precisely because his character has something irrational about it, something that reveals deep truths about human nature, not in spite of, but because of its inconsistencies and paradoxes.
Hamlet says he loves Ophelia, but he is abusive to her. This could be (and has often been so interpreted) because he is carrying out his elaborate ruse of appearing insane, but there's no proof that his behavior does not simply stem from cruelty. At Ophelia's funeral, Hamlet goes ballistic and attacks Laertes, absurdly claiming that "forty thousand brothers" could not have loved Ophelia as much as he, Hamlet, did.
Though Hamlet's obsession is to avenge his father's death, his attitude toward the Ghost is curious. He pledges vengeance but seems to mock the Ghost, calling him an "old mole." Hamlet's mindset, throughout the entire play, seems to be one of agonized self-reflection, coupled with contempt for the outside world and for the established order. He is abusive to his mother as well as to Ophelia. He evidently feels no remorse over accidentally killing Polonius. This last fact is especially interesting. Traditionally, Polonius has been portrayed as a basically harmless and rather silly old man. In Branagh's film the part is acted in a strikingly different way by Richard Briers. Not only does Polonius seem nasty and cynical, but also, one can't help thinking that he harbors incestuous feelings for Ophelia when he tries to "comfort" her after Hamlet screams at her, "To a nunnery, go!" So perhaps Hamlet's hostility to Polonius is justified, though in most interpretations, this is not the case.
All of the above would suggest that Hamlet is largely a very negative person, with a kind of boiling rage directed both inwardly and outwardly. He is clearly narcissistic and even paranoid. One wonders if he wishes to kill Claudius not just because Claudius has married Gertrude, but because Hamlet has a generalized hatred of all people. At times, Claudius, in spite of having committed murder, arguably has more claim on our sympathy than Hamlet does, not only in the scene where Claudius is praying, but also earlier, when Hamlet is taunting him, saying, "I eat the air, promise cramm'd. You cannot feed capons so." Claudius is befuddled, and honestly responds, "I have nothing with this answer, Hamlet. These words are not mine."
If, then, Hamlet has so many negative qualities, why is it that it's absolutely unthinkable to consider him a villain like Iago, or even an antihero or hero-villain, like Macbeth? I think there are two principal reasons. Hamlet is clearly a man who is suffering. In John Gielgud's recorded performance, the words "To a nunnery, go!" are pronounced with such agony that we can see, or hear, that Hamlet is dwelling in a kind of inferno. Hamlet cannot understand what is happening to him, either because the death of his father has driven him mad, or because his personality is already dysfunctional and pathological. Yet, his chief character trait, in my opinion, is alienation.
Though Hamlet views himself a stranger in the world, the words Shakespeare gives him arouse our sympathies at critical moments in the tragedy. "To be or not to be" is a soliloquy so powerfully expressive of the universal human condition that, arguably, nothing else in literature equals it. No one can listen to these words and not feel that the character who speaks them is a man who represents, and speaks for, all of us.
In summary, Hamlet is a man of largely negative outward traits. But his inner soul is in turmoil, and he is therefore an emblem of humanity. And his loyalty to his father's memory is perhaps a positive trait that cancels out the negative ones. His inconsistency and irrationality make sense, from the standpoint of drama, because these are characteristics, shown by Shakespeare in a larger-than-life form, that all people, everywhere, can understand and can identify with.