Answer:
Hamlet requested the actors to perform the play The Murder of Gonzago because it would be the perfect opportunity for Hamlet to gauge Claudius's reaction to a fictitious murder tailored to echo the murder of King Hamlet.
The lines Hamlet inserts are lines meant for Claudius's ears, implicating him as the murderer of Hamlet's father. Hamlet wants to watch Claudius's reaction and see if he shows guilt when the lines are recited.
