First point - and the most important - is that Claudius admits that he is guilty of Old Hamlet's murder. Though we are now fairly sure (just like Hamlet) that the ghost was telling the truth, Claudius confirms that his conscience is suffering. He feels guilty, he says: Polonius suggests that we can even sugar over the Devil himself, and Claudius responds
O, 'tis too true!
How smart a lash that speech doth give my conscience!
Then, Claudius goes on to foreshadow something which Hamlet will say later in the same scene to Ophelia:
The harlot's cheek, beautied with plastering art,
Is not more ugly to the thing that helps it
Than is my deed to my most painted word.
If you a compare a prostitute's naked cheek, to the cheek when it is covered in make-up, the naked one looks a lot worse. But it isn't more ugly than what Claudius has done, when you compare it to the way he sugars over it with false words. He's comparing words to deeds by virtue of made-up-women against made-up-women.
And he finishes with another reminder that Claudius is a real character, that he too feels guilty. He isn't just a baddie: he's a man with a conscience who is suffering real guilt:
O heavy burden!