In a revenge play, almost always the revenger dies, or is killed, sometimes surrendering himself to the inevitable - hence 'revenge tragedy' - and I would argue that Hamlet is this as well. But nothing is simple in this play.
However, the 'revelation' of tragedy occurs far earlier than at the end. If we take 'tragedy' to be the dramatic working out of a hero, his fatal mistake (sometimes considered to be his tragic 'flaw' - the weakness in his moral nature that causes the mistake) and the consequences of that, then Hamlet's accidental killing of Polonius is the moment at which Hamlet becomes a murderer. Moreover, he has killed the wrong man, believing the figure behind the arras in his mother's chamber to be the villain, Claudius, whose death would have have been justified in purely 'revenge' terms: instigated by the Ghost urging his son to 'remember' him by avenging his death.
The moment at which Hamlet becomes a killer is certainly a point of no return in the play - from this point on, we can only expect dire consequences and a tragic ending. But there are surely intimations of tragedy even before this: the portents of the king's Ghost on the eve of war; the new king, who has ostensibly killed his brother and stolen his kingdom and his queen, Hamlet's madness, which might or might not be 'craft', together with his debates on the meaning of existence, human and general, as well as his own, all point in the tragic direction from the outset. The ending is both heroic and tragic. It is also (consider Gertrude's drinking poison, Laertes dying of poisoned wounds meant for Hamlet, Horatio being dissuaded from suicide in order to tell Hamlet's story) as highly ambiguous as almost everything else in this most complex of dramas.
Some critics consider Hamlet to be a revenge play, but it is perhaps in the final scene where the real nature of it as a tragedy is revealed. As Hamlet considers his possible courses of action throughout the play, the nature of his revenge is a major consideration. He cannot kill Claudius while he is praying for he will then be sent straight to heaven. And he worries about his mother's reaction as well.
But in the final scene, where Claudius is killed and Gertrude dies along with Hamlet and so many others, the fatal flaw of Hamlet is shown again and the fact that he dies while achieving his revenge makes it clear that it is a tragedy and not a revenge play.