What has Hamlet done with the play, and what is the play titled?

The play in entitled, "The Mousetrap". Hamlet has changed the play to reflect or reenact his father's murder.

The play within a play is two scenes. It has a dual purpose. The second scene is taken from "The Murder of Gonzago". Hamlet will use it to catch the conscience of the king. The first scene, between the Player King and the Player Queen is a mousetrap Hamlet will use to catch the conscience of the Queen. If indeed she was involved in the murder of her husband, King Hamlet. It's called "The Mousetrap" presumably because "Mouse" is a pet name Claudius uses to refer to Gertrude according to Hamlet a few scenes later.