There are probably many different hamartia's exibited by Hamlet, the one that seems most obvious would have to be his indecisivness. Hamlet seems to have trouble making up his mind about a couple of different dilemma's that he is faced with. A good example is his struggle with wether or not he should kill Claudius. Hamlet's flaw was that he ruined his life by not confronting the problem earlier, instead he wanted to obtain proof beyond a reasonable doubt to justify what he wanted to do to Claudius, and nothing less would suffice. Hamlet not only wanted to kill Claudius, he wanted to damn him to hell. However, during his mental battle he wasn't seeing the damage he was doing to himself. By Feigning madness he almost got himself killed a few times, he destroyed the relationship he had with his mother, and he sent Ophelia into a downward spiral of depression that ended with her suicide, and her suicide led to to the final bloodbath at the end which saw nearly everyone killed. His fake insanity was his indecisivness in action, he decided to act insane so he could hide his motives, but other than that he didn't know what he was going to do. He didn't have the stomach to ascend the throne by exposing his uncle and then killing him. Although he did end up killing Claudius at the end, it was only because of the madness that went on in the final scene. Had he not been challenged at the end, I don't believe he would have killed Claudius, regardless of the evidence he obtained against him.
Claudius' only hamartia is his cold, calculating, evil nature. Claudius causes his own ruin because of his evil behavior. He murders the queen because he has a lust for power, and perhaps for Gertrude, he spied on Hamlet and concocted several plans for his murder. His death would have been avoided if he didn't have the kind of prsonality that allows him to kill everyone in the way of getting what he wants.