Hamlet tells Claudius to search for Polonius in hell because of what

First Hamlet instructs Claudius to send a messenger to heaven to check out that Polonius is not there. (He likely isn't) If Polonius is not in heaven, there is only one other place he could be and that is in hell. There could be lots of choices. The obvious one is that Polonius is dead, and he has to be somewhere. Hamlet jokes about the death a bit. Polonius is not his favorite person. Added to that, Hamlet mistakenly believes that he has killed Clausius. Hamlet was not happy to find out that he hadn't.

To announce his death he says that there's no need to hurry, Polonius is not going anywhere. Claudius sends Hamlet to England and then informs England (the king) that he wants Hamlet dead. The plot is convoluted but the poetry is astonishing and the characterization second to none.