How are the two kings, Hamlet's dead father and Claudius, contrasted with one another throughout the play?

In addition to the superb answer above, King Hamlet
& the Ghost can be juxtaposed with Claudius in terms of their own language and
actions and not just on Hamlet's obviously biased
soliloquy.

Both have killed to gain or
maintain power. King Hamlet killed Old Fortinbras to gain lands in
Poland.  Claudius obviously kills King Hamlet to get the
throne.

Both loved Gertrude.
The Ghost tells Hamlet to leave her to heaven, and Claudius tries to get her to stop
drinking the poisoned chalice.

Both are
blind. King Hamlet was blind to his wife and brother's incest and
adultery.  Claudius is blind toward Hamlet and the effects of his incestuous and
murderous crimes on the state.

Both form
unnatural relationships. The Ghost, a supernatural being, elicits a
mortal (his son, no less) to carry out what he should do (haunt Claudius).  Claudius'
relationship with Gertrude is full of incest, adultery,
deceit.

Both want revenge.
King Hamlet wants revenge on Claudius, and Claudius wants revenge on Hamlet.  They both
incite Hamlet to enact revenge.

Both use
pawns as part of revenge. The Ghost uses Hamlet to carry out his plan,
while Claudius uses Polonius' family, R & G.  All of their pawns, by the way,
die.

King Hamlet (Ghost) will go to heaven,
and Claudius will go to hell. The former is a victim of immorality, while
the former is an agent of it.

The Ghost is a
better performer than Claudius. His lines are war-like, gothic,
menacing.  Claudius, especially at prayer, seems much weaker by
comparison.