Why doesn't Hamlet show guilt after he kills Polonius?
Hamlet has just begun to express his feelings to his mother about her marriage and her intimate relations with Claudius, which he considers disgusting, adulterous, incestuous, and wicked. It is impossible for him to add any additional feelings to those which have already driven him half mad. His anger frightens his mother so badly that she cries for help. Then Polonius, who is behind him and hidden behind a tapestry, begins calling for help. Hamlet is not only consumed with anger but confused. He suspects that he has walked into a trap, because both his mother and Polonius are calling for the guards. If the guards arrest him he will be put in a locked room, or possibly even into a dungeon. He may lose his liberty forever, and be at the mercy of his enemy King Claudius. When he kills Polonius it is impossible for him to add any more feelings to those he is already experiencing, which include pent-up anger, disgust, hatred of Claudius, suspicion, fear of arrest, and fear for his own life.
He does not understand the truth of the situation. His mother thinks he is insane. He grabs her to prevent her from leaving and tells her he intends to set her up a glass (mirror) where she can see the inmost part of herself. She takes this to mean that he literally intends to cut her open with his sword and expose her inner organs. She has already threatened to send for assistance, and now she starts screaming. Polonius can't tell what is going on because he is covered by the arras, but he assumes that Hamlet is murdering his own mother. Both Gertrude and Polonius think Hamlet is insane. They know nothing about the Ghost or what the Ghost has told Hamlet or why Hamlet has decided to pretend to be crazy.
Hamlet decided to pretend to be insane because he understood that Claudius was suspicious of him and was trying to figure out--as well as to get others to help him figure out--what was going on inside his mind (or soul). Claudius is trying to get Polonius, Gertrude, Ophelia, Rosencrantz and Guildenstern all to help him read Hamlet's mind. Hamlet thought he could hide his real thoughts, feelings, and intentions from the King's prying eyes by making him believe he was insane. No one can understand a person who is insane. The only way to pretend to be insane effectively is actually to become a little insane.
Hamlet had further reasons for not showing guilt after killing Polonius. The old man was spying on him and plotting against him. The old man was a trouble-maker. The old man had interfered in his relationship with Ophelia. The old man was turning Hamlet's own mother against him. The old man was an enemy who could do him a lot of harm because he was close advisor to the King. This is why Hamlet says, "heaven hath pleased it so, / To punish me with this and this with me." Hamlet feels such remorse that he feels punished by heaven, yet he sees such harm being done by Polonius, that he feels Polonius is punished, at Hamlet's own hand, by heaven, too.
Hamlet’s immediate reaction to the killing of Polonius is to wonder who he has killed.
At this point, Hamlet has been wanting to kill King Claudius for a while. When he stabs the unseen intruder, he doesn’t yet know who it is.
Here are the words Hamlet and his mother, Queen Gertrude, exchange just after the killing:
O me, what hast thou done?
Nay, I know not:
Is it the king?
Notice that Hamlet asks if it’s the king. For a moment he believes that it’s possible that he’s actually carried out the act that he has been so conflicted about. When he finds out that it’s Polonius, he is most likely disappointed.
However, Hamlet is not entirely without remorse. Late in the scene he does say he’s sorry for what he’s done:
For this same lord,
[Pointing to POLONIUS]
I do repent: but heaven hath pleased it so,
To punish me with this and this with me
This also isn’t the first time that Hamlet has shown a lack of feeling. He also told Ophelia that he did not love her earlier in the play, after behaving otherwise. We can conclude that Hamlet is so distracted and consumed with avenging his father’s death that he is not in touch with his own feelings.