In my opinion there was no NEMESIS in Hamlet. The downfall of Hamlet does not occur due to fate the actions all could have been avoided by the choices of either Hamlet or others in the play.
These lines show the ANAGNORISIS of Hamlet, because he realizes his actions led to his defeat and he knows that Fortinbras acted as he himself should have, causing Hamlet to vote him for king.
The poison is killing me, I will not make it to hear what has occurred in England. But my guess is Fortinbras will be chosen as king and he has my vote (5.2.390-393)
These lines are the CARTHARSIS of the play. Because we as the audience begin to pity Hamlet as he works to keep Horatio alive. We fear for him not wanting another person he loves to die becasue of him. We begin to feel this way towards him because while he hasnt fully shown he knows the error in his ways he is beginning to.
Give the cup to me! I will have it. Horatio if you ever cared about me, you must live, and tell my story(5.2.378-383).