Claudius murdered Hamlet's father (the former king of Denmark), married the dead king's wife, Gertrude—with whom he was probably already having an affair—and usurped the throne from the dead king's son, Hamlet, who was at school in Germany and oblivious to everything that was going on back home in Denmark.
Claudius doesn't regret killing the old king, and he's not particularly remorseful about it.
CLAUDIUS. O, my offence is rank, it smells to heaven;
It hath the primal eldest curse upon't,
A brother's murder! Pray can I not,
Though inclination be as sharp as will;
My stronger guilt defeats my strong intent... (3.3.38–42)
In a moment of weakness after commissioning Rosencrantz and Guildenstern to take Hamlet to England to be killed, Claudius decides that it might be advantageous to his soul to ask God for forgiveness for the murder of old King Hamlet—and possibly for ordering Hamlet's murder as well.
Claudius knows that he's probably going to hell for the murder of the old king. What's done is done, he says, and there's really nothing he can do about it now, but he rationalizes to himself that it can't hurt to ask.
CLAUDIUS. And what's in prayer but this twofold force,
To be forestalled ere we come to fall,
Or pardon'd being down? Then I'll look up;
My fault is past. (3.3.50–53)
Claudius is almost mocking of himself in this soliloquy. "What do I say?" He says to himself. "Ha! What can I say?"
CLAUDIUS. But, O, what form of prayer
Can serve my turn? "Forgive me my foul murder?" (3.3. 54–55)
Claudius is far too realistic and pragmatic to believe that kind of prayer will actually work with God.
CLAUDIUS. That cannot be; since I am still possess'd
Of those effects for which I did the murder—
My crown, mine own ambition, and my queen.
May one be pardon'd and retain the offence? (3.3.55–58)
Claudius has made no attempt whatsoever to repent his sins, and he has no intention of offering any kind of compensation to his victims. He's already married one of them (the dead king's wife, Gertrude) and he's not going to relinquish the crown to Hamlet, the old king's rightful successor. "Can I keep all these things and still be forgiven?" he asks. "What about if I can't really repent for my sins?"
CLAUDIUS. What then? What rests?
Try what repentance can. What can it not?
Yet what can it when one cannot repent? (3.3.66–68)
Nevertheless, Claudius decides that it's worth a try, and he thinks that if he gets on his knees and acts repentant enough, God might just forgive him.
CLAUDIUS. Help, angels! Make assay.
Bow, stubborn knees; and heart with strings of steel,
Be soft as sinews of the new-born babe!
All may be well. (3.3.71–75)
Claudius ultimately decides that there's nothing he can say or do that will encourage God to forgive him. The simple reason is that Claudius has no regrets, feels no guilt, and doesn't really want to repent for what he's done to the old king or for what he intends to do to Hamlet.
KING. My words fly up, my thoughts remain below.
Words without thoughts never to heaven go. (3.3.99–100)