How are the themes of revenge, nemesis, and death interrelated in Shakespeare's Hamlet?

Hamlet wants to get revenge on his nemesis, Claudius, for killing his father.

A nemesis is an enemy, but it is usually used to refer to an enemy of a personal nature.  When Hamlet learns that his father was killed by Claudius, he is horrified.  This information comes from his father’s ghost.

Ghost

Revenge his foul and most unnatural murder.

HAMLET

Murder!

Ghost

Murder most foul, as in the best it is;
But this most foul, strange and unnatural.

HAMLET

Haste me to know't, that I, with wings as swift
As meditation or the thoughts of love,
May sweep to my revenge. (Act 1, Scene 5)

Hamlet’s ghost wants Hamlet to avenge his death.  Hamlet is already angry at his uncle for marrying his mother.  It is even worse to learn that he accomplished that by killing Hamlet’s father.  The idea of killing him in revenge makes sense.

Hamlet has a little bit of a hard time getting revenge for his father.  He concocts an elaborate plan, which involves Hamlet pretending he is crazy and trying to get his uncle to incriminate himself by staging a play with a similar plot to the murder of Hamlet’s father.

The spirit that I have seen
May be the devil: and the devil hath power
To assume a pleasing shape; yea, and perhaps
Out of my weakness and my melancholy,
As he is very potent with such spirits,
Abuses me to damn me: I'll have grounds
More relative than this: the play 's the thing
Wherein I'll catch the conscience of the king. (Act 2, Scene 2)

That doesn’t really work, if Hamlet expected Claudius to jump up and confess.  Eventually, Hamlet does get revenge on his nemesis, but at great cost.  In a duel with Ophelia’s brother, Hamlet is killed along with his uncle and his mother.  Everyone is poisoned, either by the sword or the drink.  Nothing goes quite as planned, but Hamlet's father is avenged.