In Hamlet, how does Claudius respond to the death of Polonius? Does he understand the implications of what happened?
Claudius responds to the death of Polonius by sending Hamlet to England, along with secret orders to have him killed. He fully understands the implications of what has happened. He understands that when Polonius hid behind the arras to spy on Hamlet during Hamlet's conversation with his mother, Hamlet thought it was Claudius hiding there. When Hamlet stabs Polonius, he thinks he is stabbing Claudius: Claudius recognizes quite clearly that Hamlet was trying to kill him, not Polonius.
Having seen the mousetrap play, Claudius knows that Hamlet has discovered that he, Claudius, murdered Hamlet's father. He now knows that Hamlet will act boldly to avenge his father's death. He knows that his life is in acute danger and that Hamlet has moved from despondent depression to violent action. It's imperative that he get Hamlet away from the Danish court as quickly as possible. He knows he has to get Hamlet out of Denmark rather than jail him for Polonius's murder because Hamlet is popular and if the people want it, the young prince could easily be crowned, deposing Claudius. At this point the die has been cast: Claudius realizes that either he or Hamlet will soon be dead and tries to ensure that the corpse will be Hamlet's. As Claudius puts it,
How dangerous is it that this man goes loose! ...
Diseases desperate grown
By desperate appliance are relieved,
Or not at all.
Yes, in Shakespeare's Hamlet, Claudius certainly understands the implications of Hamlet's killing Polonius, as well as the implications of the situation as a whole. When he first hears the news that Hamlet has slain Polonius, he tells Gertrude:
His liberty is full of threats to all--
To you yourself, to us, to everyone. (Act IV:1)
Later, he adds:
How dangerous it is that this man goes loose! (IV:3)
Claudius is preparing Gertrude for what he is about to do to counter Hamlet's threat to the king's thrown and life. Yet, he also understands that he must be careful concerning what he does to Hamlet, due to his popularity with the Danish people:
Yet must not we put the strong law on him.
He's loved of the distracted multitude,... (IV:3)
Publicly, then, Claudius orders Hamlet sent to England (even though at the beginning of the play he wanted him to stay in Elsinore so that he could keep an eye on him). Secretly, he arranges for Hamlet to be put to death upon his arrival in England.
Remember, this is after the performance by the players, and thus after Claudius knows that Hamlet knows about the death of Hamlet's father. Claudius knows what's going on, and takes steps to counter the threat. He is a formidable adversary. Polonious is an idiot and a blowhard, but Claudius is no fool.