After the story of Priam's slaughter, what does Hamlet ask the actors to do and why?

After everyone departs, Hamlet holds back the First Player
and asks him if he can play the Murder of Gonzago. When he acknowledges that he can,
Hamlet directs that it be presented "tomorrow night". He then asks if the First Player
could study some lines that would be inserted into the play. The First Player accepts
that as well.

As we shall see in the soliloquy to follow,
Hamlet is going to use this little production to catch the conscience of the King. But
it is more than that. The twist he is putting into the play of the additional lines will
turn the play into "The Mousetrap". Foremost in his mind is his mother's fall from grace
and he intends to catch her too. If, that is, she had a part in the poisoning of King
Hamlet.

When the players and audience are assembled in 3.2,
the whole first act is taken up with probing the conscience of the Queen. She appears to
pass the test. When the Murder of Gonzago appears however, the king does rise and leaves
the play. It would seem Hamlet has caught a rat! The irony of course is that Hamlet has
run his mouth a little too much at the end and given the audience the notion that the
play is about a nephew killing the King.