Claudius does not ask Hamlet what he has done with Polonius, although Rosencrantz and Guildenstern do. He asks Hamlet where Polonius is. His lines are "Now, Hamlet, where's Polonius?" and "Where is Polonius?". Hamlet answers, first that Polonius is at supper (not where he eats but where he is eaten) and second that Polonius is in heaven (where Claudius cannot go to find him). Then he adds, "But if indeed you find him not within this month you shall nose him as you go up the stairs into the lobby."
hamlet killed polonius by accident. polonius was hiding behind the arrays in gertrudes room and hamlet thought it was claudius so he stabbed him with his sword only to find that it wasnt cladius, but polonius.
I think he was afraid Hamlet was going to find out that Claudius killed Hamlet's father.
As Claudius's advisor, it was Polonius's job to advise him on all matters of state. We presume that Polonius advised Claudius to deal with Fortinbras diplomatically through his uncle the king of Norway. This appears to have been successful. He also advised Claudius that in his opinion Hamlet was suffering from unrequited love for Ophelia and the best way to find out what was going on with Hamlet was to throw everyone Hamlet appears to have cared about at him, including Ophelia, Rosencrantz, Guildenstern and Gertrude, and spy on the conversation. This was not such good advice.
Hamlet doubts the ghost's statement that Claudius had murdered him--the ghost could be lying. He re-enacts the murder in a play and has it performed in front of Claudius. Sure enough, it causes an attack of conscience in Claudius.
Hamlet's uncle Claudius killed Hamlet's father (called Hamlet Sr.). The ghost of Hamlet Sr. comes back from the dead and tells Hamlet Jr. so. Hamlet Jr. confirms this by re-enacting the murder as a play, which upsets Uncle Claudius.
As he is dying, Laertes tells Hamlet.
Hamlet tells him to send his messengers to see if Polonius is in Heaven, and if he is not there, then he tells him to go see if he's in Hell in person. He then tells him that Polonius is under the stairwell.
Polonius sends Reynaldo to spy on Laertes; Ophelia reports that Hamlet behaved weirdly in her bedroom; the King and Queen hire Rosencrantz and Guildenstern to find out what's up with Hamlet; Polonius tells the King and Queen his theory about Hamlet's madness; Polonius gets shown up as a fool by Hamlet; Rosencrantz and Guildenstern meet with Hamlet; he guesses their purpose and gives them the "What a piece of work" speech; the actors arrive and the main actor performs a speech of Aeneus' tale to Dido about the fall of Troy; Hamlet on his own delivers the "O what a rogue and peasant slave am I" soliloquy and decides to use the actors to reveal Claudius's guilt.
Polonius is one of those people who, once he gets an idea, there's no shifting it. He has formed the idea that Hamlet is mad for Ophelia's love, and suggests that this can be proven by spying on the two of them from a concealed place. Claudius is not persuaded at all by this: "Love? His affections do not that way tend." But Polonius has a new idea (which is basically the same as the old one) that his ideas may be proven by spying on Hamlet's conversation with Gertrude from a concealed place. Claudius just wants to get rid of him by sending him to England. Polonius therefore views the conversation with Gertrude as his last chance to prove his theory. "If she find him not, to England send him, or confine him where your wisdom best may think."
Polonius explains this earlier in the play, in 2.2. P: You know sometimes he walks four hours together Here in the lobby. Q: So he does indeed. P: At such a time I'll loose my daughter to him Be you and I behind an arras then. Mark the encounter. If he love her not, And be not from his reason fall'n thereon, Let me be no assistant for a state But keep a farm and carter's.
Claudius fears that his people would end up supporting Hamlet should he have remained in Denmark. Thus, Claudius decides to send Hamlet off. Basically, it was a public relations mess for Claudius since he had to find the right balance of punishment and leniency.