Hamlet kills Polonius by mistake, but he feels no guilt for killing him. After staging the "Mousetrap," or the play with in the play, to test his theory that Claudius planned and carried out the king's death. After Claudius runs screaming for light, Hamlet follows his mother to her confront her about what she has done.
Polonius tells Gertrude to talk with him, and that while they speak, he will hide behind the tapestry. He wants her to speak with him about his behavior and find a way to calm him down.
He will come straight. Look you lay home to him.
Tell him his pranks have been too broad to bear with,
And that your Grace hath screen'd and stood between
Much heat and him. I'll silence me even here.
Pray you, be round with him.
As their conversation escalates, Gertrude becomes afraid. Her son is impassioned and yelling at her by bringing a mirror to his actions: marrying her husband's brother and her son's uncle.
What wilt thou do? Thou wilt not murder me?
Help, help, ho!
Polonius is also afraid for Gertrude and calls out for help, "What, ho? Help, help, help!" Thinking that it is Claudius (or a rat "How now, a rat? Dead for a ducat, dead!") Hamlet drives his sword through the tapestry and into Polonius.
Gertrude is understandable distraught that her son has so callously murdered someone. "O me, what hast thou done?" she asks him incredulously. In an almost flippant tone, Hamlet responds that he doesn't know. Did he kill the king? Oh, well it wasn't him. At least killing a king wasn't as bad as what she's done.
A bloody deed? Almost as bad, good mother,
As kill a king and marry with his brother.
In the next act, Hamlet plays a game of hide and go seek with Claudius and his men as they try to locate where he has hidden Polonius' body. With such word play, clearly he doesn't feel bad about killing Polonius.
Take you me for a sponge, my lord?
Ay, sir; that soaks up the King's countenance, his rewards, his authorities. But such officers do the King best service in the end. He keeps them, like an ape, in the corner of his jaw; first mouth'd, to be last swallowed. When he needs what you have glean'd, it is but squeezing you and, sponge, you shall be dry again.