If by "redeem" himself you are asking how does Hamlet finally complete the vengeance that was asked of him by the ghost of his father in Act 1, then you need only look at the fact that he directly kills Claudius in Act 5 scene 2, not only for killing his father, but also for his responsibility in the death of Gertrude and Hamlet himself. Hamlet learns that he has been mortally wounded, but his final acts are what the entire play has been building toward. He doesn't have a specific plan to kill Claudius in this final scene; he only knows that he is being asked to participate in a fencing match with Laertes, but that doesn't mean that he is a fool walking into his own death. He knows that Claudius is out to kill him and that Laertes could be dangerous in his own desire for vengeance for Polonius's death. Hamlet just realizes though that "the readiness is all." All he can do is be ready for what may come and do the best he can in whatever circumstances he finds himself. He finds himself with further proof of Claudius's treachery and a sharpened and poisoned sword -- and he uses them against Claudius! If you see Hamlet's cautious behavior as a fault, then these actions could be considered his redemption.