How did Hamlet find ways to make Claudius guilty about Hamlet Sr.'s death?

At the end of act 2, Hamlet declares that "the play's the thing / wherein I'll catch the conscience of the king." This refers to a play he plans to have performed in front of Claudius, in which the players will act out "something like the murder" of Hamlet's father. Hamlet hopes that Claudius will be made to feel so guilty and so "struck so to the soul" that he will then confess. Appropriately, Hamlet decides to call the play "The Mouse-trap."
In act 3, scene 2s, just before the play is about to begin, Hamlet instructs Horatio to observe Claudius. He says that if Claudius's "occulted guilt" does not "itself unkennel in one speech," then the ghost they have seen, and which told them that Claudius murdered the previous king, must be "a damned ghost." Thus Hamlet puts on this play not simply to make Claudius feel guilty, but to reassure himself that Claudius really is guilty.
Hamlet's plan seems to work. When in the play the murderer pours poison into the sleeping victim's ear, Claudius rises to leave. This is, of course, the same method he used to kill the previous king, Hamlet's father. As the king rises, startled, Hamlet comments, contemptuously, "What, frighted with false fire!" At this moment, the king's guilt is confirmed.