When Act 3, Scene 3 opens, Claudius is expressing his concerns and intentions to Rosencrantz and Guildenstern:
I like him not, nor stands it safe with us
To let his madness range. Therefore prepare you;
I your commission will forthwith dispatch,
And he to England shall along with you:
The terms of our estate may not endure
Hazard so dangerous as doth hourly grow
Out of his lunacies.
Claudius has been severely shaken by what happened in the previous scene. Hamlet staged a play in which a murder was enacted that was a close copy of what the Ghost had described to Hamlet as having been done to him by Claudius. Several good questions might be asked regarding these opening lines spoken by the King.
Does Claudius believe that Hamlet is insane, or is he afraid that Hamlet is only too sane and has somehow learned that he murdered his own brother in order to usurp the throne? How could Hamlet know about the way the crime was committed?
Nobody else in the audience was alarmed when they saw the actor pouring poison into the sleeping player-king's ear. Only Claudius--because he has a guilty secret. Would anybody agree with Claudius that this was intended as a threat to him, and that that was why he fled the room?
How can what happened in the play-within-a-play be construed as an act of madness on Hamlet's part? Hamlet didn't write the play. He only chose to have that particular play, which he calls "The Mousetrap," presented. What if he had staged a play in which somebody murdered a king with a dagger while he was in bed?
In the opening lines of Act 3, Scene 3, Claudius gives orders for Hamlet to accompany Rosencrantz and Guildenstern to England. At this point, is Claudius already planning to have Hamlet beheaded by the English, or does he make that decision after Hamlet kills Polonius in Act 3, Scene 4?
Act 3, Scene 3 of Hamlet is pivotal because it establishes that Claudius is sending Hamlet to England, and that Polonius is going to hide behind the arras in Gertrude's chamber, where he will be killed in the following scene.
Does Gertrude believe her son is mad because of what happened at the play-within-a-play? What exactly did happen? Did the entire audience get the idea that Hamlet was deliberately accusing Claudius of murdering his brother by staging a play in which a similar murder is perpetrated? Even if the audience at the play, including Gertrude, believed that Hamlet was suggesting that Claudius murdered King Hamlet, how could anyone other than Claudius himself sense that the assassination was committed by the highly unusual method of pouring poison in a man's ear while he was asleep?
Gertrude screams for the guards because she thinks her son is insane. What has made her think that? When did she begin to think it? In Act 3, Scene 4, Hamlet grabs her arm and tells her:
Come, come, and sit you down; you shall not budge;
You go not till I set you up a glass
Where you may see the inmost part of you.
Hamlet is speaking metaphorically--but the Queen thinks he means it literally. She thinks in his madness he intends to cut her open with his sword and make her look at her exposed internal organs in a mirror. This is stroke of genius! She screams for help.
Polonius can't see a thing where he is hiding, so he starts shouting. Hamlet is shocked and confused. He thinks he has walked into a trap. If the guards arrest him, he may be kept in confinement for life, if not murdered.