Claudius sees that Gertrude is upset. She says Hamlet was acting insane, and in his madness killed Polonius.
It's unclear if Gertrude is keeping Hamlet's secret or really does think he's mad.
Claudius exclaims that if he had been behind the tapestry, he would now be dead. He thinks of how best to explain the murder to the public, and sends Rosencrantz and Guildenstern to find Hamlet.
Claudius doesn't mourn for Polonius, instead worrying about himself and how to manage the murder politically.