In "Hamlet", when and to whom does Claudius openly reveal his determination to remain king?

Ironically, Claudius is praying to God in the aftermath of "The Murder of Gonzago" when, thinking no one is listening, he confesses he intends to remain king. He has just conspired to rid himself of Hamlet by sending Hamlet to England along with Rosencrantz and Guildenstern. As we later discover, Claudius will send a note to the King of England asking England's king to kill Hamlet. He asks God for some kind of forgiveness but admits:

"O, what form of prayer

Can serve my turn? 'Forgive me my foul murder?'

That cannot be; since I am still possess'd(55)

Of those effects for which I did the murder—

My crown, mine own ambition, and my queen."

( Act 3, Scene 3)

What Claudius does not realize is that Hamlet is listening to Claudius and planning to kill him then and there. So, Claudius has actually confessed his actions to Hamlet, as well as God. However, Hamlet refuses to kill him, because Claudius is praying and, according to religious belief at the time, being killed by praying would allow Claudius to go directly to heaven while his father "died in his sins" and is still in purgatory. Once again, Shakespeare shows the irony of the situation, because after Hamlet leaves, Claudius admits he really can't ask for forgiveness for something which he is not sorry. He states,

"My words fly up, my thoughts remain below.

Words without thoughts never to heaven go."(100)

Thus Hamlet has misses a great chance to kill Claudius.