In "Hamlet", when and to whom does Claudius reveal his determination to remain king of Denmark?
One could argue that Claudius reveals his determination to remain King when he orders Rosencrantz & Guildenstern to take Hamlet to England. He explains to them, "I like him [Hamlet] no, nor stands it safe with us / To let his madness range" (3.3.1-2) because he believes, "The terms of our estate may not endure / Hazard so near's as doth hourly grow / out of his brows" (3.3.5-7). In the same scene, Claudius prays to God, he questions, "My fault is past. But, O, what form of prayer / Can serve my turn? 'Forgive me my foul murder'? / That cannot be, since I am still possessed / of those effects for which I did the murder-- / My crown, mine own ambition, and my queen" (3.3.51-55). He believes hecan and will retain his Kingdom since his offense is in the past/is history. He reasons that he does not need to ask for forgiveness since he still retains his crown and the queen. Therefore, even though he proclaims his intention only to himself, his prayer still reveals his determination to remain King.
Perhaps one other example involves the entire plan for Laertes to avenge Polonius' death by fighting and killing Hamlet in Act V (They make the plan in Act IV.). He even decides to poison the cup of wine as a backup plan in case Laertes is unsuccessful (4.7.157-160). If he can rid himself of Hamlet, no one will ever know he murdered his brother.