CLAUDIUS: 'Tis sweet and commendable in your nature, Hamlet,
To give these mourning duties to your father:
But, you must know, your father lost a father;
That father lost, lost his, and the survivor bound
In filial obligation for some term
To do obsequious sorrow: but to persever
In obstinate condolement is a course
Of impious stubbornness ....
-- Hamlet, I.ii
I must have read or seen Hamlet dozens of times before I grasped one of the implications of this passage. I've always seen the obvious one, that Claudius is attempting to distract Hamlet from his grief. (He goes on to literally promise him the kingdom, for Pete's sake.) It's not hard to see why Claudius, the murderer, doesn't want anyone -- least of all Hamlet -- dwelling on the murder of Hamlet's father. Move along, Danes; nothing to see here!
What I noticed only recently is something Claudius himself doesn't mean to reveal: his own casual, brutal attitude toward death. When he says airily that "your father lost a father," he as Hamlet's paternal uncle is speaking of his own father's death. And it means nothing to him. Death has never meant anything to Claudius, except lately as a means to an end.
More deeply, he's unconscious that death could mean something to other people. He's unaware he's revealing anything about himself because he's unaware that he's unusual -- practically sociopathic. No doubt the way he counsels his nephew Hamlet to experience the elder Hamlet's death, and the subsequent natural grieving process, is simply the way Claudius does it himself: you put on some black clothes, walk around pretending to look sad for a while, and then have a nice sandwich.
He's manipulating Hamlet, to be sure, but I think he's also genuinely puzzled by Hamlet's behavior. Why are you acting this way, kid? Don't you know this isn't how you play the game?
Claudius is creepy. Claudius is cold.