Like Macbeth, Claudius is aware of his crimes without being able to repent of them. Both have chosen to damn themselves for worldly advantage, but are too proud or too despairing to change their ways.
In Act III of Hamlet, Claudius is in private agony throughout, though he conceals it. Upon hearing Polonius make a chance remark on hypocrisy in Act III Scene 1, he mutters,
O, 'tis too true!
How smart a lash that speech doth give my conscience!
O heavy burden!
In Act III, Scene 2, Claudius cannot bear to see the end of the play that Hamlet has arranged for him, which alludes to the way in which Claudius has murdered Hamlet's father, but rushes off in a panic: "Give me some light:--away!" He pretends to his attendants that what has affected him is "choler" or anger, but we understand clearly enough that it is his conscience.
In Act III Scene 3, Claudius is again overcome with guilt after putting on a decisive face before others. Following a speech that begins
O, my offence is rank, it smells to heaven;
It hath the primal eldest curse upon't,--
A brother's murder!...
he turns to prayer. Nevertheless, he knows that at the end his dilemma remains the same:
My words fly up, my thoughts remain below:
Words without thoughts never to heaven go.
Despite the pangs of his conscience, Claudius is unwilling to take full ownership of what he has done. In the end, like Macbeth, if he can somehow get through in this world, he will disregard his conscience.