Hamlet is given to much self debate. What does Hamlet think about in Act 5 of Hamlet?

In addition to Hamlet's sense of identity in that he is
now Prince of Denmark, after looking at the skulls of the graveyard he realizes that
death is a great equalizer.  And, with his graveyard experiences, Hamlet now has a
sense of fate--

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When our deep plot fail and that should teach us
that

There's a heavenly power that shapes our
ends,

No matter how much we think we're in
control.(5.2.8-10)

With this
new self-definition and the awareness of Fate and equality of death, Hamlet declares
that he feels no guilt about the deaths of Rosencrantz and Guildenstern who have tried
to have him killed: 

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...their defeat

Does
by their own insinuation grow.
(5.2.62-63)

Likewise, Hamlet,
Prince of Denmark, feels justified and driven by Fate now in avenging his father's
murder against Claudius who must be stopped:

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Does it not, think thee, stand me now
upon--

He that hath killed my king and shored my
mother,

Popped in between th'election and my
hopes,

Throw out his angle for my proper
life,

And with such coz'nage--is't not perfect
conscience

To quite him with this arm?  And ist' not to be
damned

To let this canker of our nature
come

In further evil?
(5.2.68-75)

Inspired further
by the noble character of Fortinbras,  the once self-debater has reached his final
resolve as the Prince of Denmark.  He must be rid of the king.  So, he accepts the
challenge and has to duel Laertes.  After this duel, Claudius is slain.  However, as he
lies weakened and dying, Hamlet reconciles with Laertes and hands over the power of
Denmark to Fortinbras, defeating the king.