He's the antagonist.
King Hamlet, in Shakespeare's play, was poisoned by his brother Claudius.
Hamlet asks Horatio to observe Claudius.
No, Claudius is Danish and the Ptolemies are Egyptian. There is nobody called Ptolemy in the play Hamlet.
Hamlet finds Claudius praying for forgiveness after the play. He decides that if he were to kill Claudius at that moment, Claudius would go to heaven rather than to hell. Hamlet decides to wait until he finds Claudius sinning, in order to kill him.
The Murder of Gonzago. Hamlet nicknames it The Mousetrap.
In Shakespeare's play Hamlet, Claudius is Hamlet's uncle and also his stepfather. There isn't a Claudia.
Basically, Hamlet asks Horatio for help in keeping an eye on Claudius during the 'Mousetrap' play.
Hamlet's uncle Claudius killed Hamlet's father (called Hamlet Sr.). The ghost of Hamlet Sr. comes back from the dead and tells Hamlet Jr. so. Hamlet Jr. confirms this by re-enacting the murder as a play, which upsets Uncle Claudius.
He doesn't. Hamlet tells Claudius that the name of the play is the Mousetrap, not the other way around. The actual name of the play is The Murder of Gonzago but Hamlet is using it to trap Claudius and so gives it a different name.
The Mouse Trap
In Shakespeare's play Hamlet, King Claudius and Polonius eavesdrop on Hamlet's "To be or not to be" soliloquy.
When Claudius flees from the 'Mousetrap' play, as Hamlet hoped he would.