Claudius's inner conflict is first expressed in his long soliloquy in Act 3, shortly after the play within a play. He is clearly unsettled by having just watched a performance of what he actually did to his brother, and his guilt is getting to him. This soliloquy takes place in a chapel in the castle, and the speech is almost an attempt at prayer.
Claudius knows that his murder of his brother and the marriage to Gertrude are unholy acts. He opens the speech by saying his "offense is rank, it smells to heaven." He goes on to say that while he IS inclined to pray, he knows he really can't -- after all, what can he say that would bring him God's forgiveness? He sarcastically suggests that he could say, "'Forgive me my foul murder.'" But he knows how ludicrous that would be, and he understands why. In the next line he states directly that he knows he can't be forgiven if he isn't willing to give up the things he has gained from the crime: "my crown, mine own ambition, and my queen." Even though he forces himself to kneel and tries to pray, the final words of the scene reveal the reality. Claudius concludes, "My word fly up, my thoughts remain below; words without thoughts never to heaven go." He can pray all he wants, but if he doesn't really mean the words, he will never be forgiven for the crimes. His conscience is in conflict with his earthly desires.
An additional conflict he faces throughout Act 4 is doing what he thinks he has to do to get rid of Hamlet, all the while pretending to love Hamlet so as to protect Gertrude's feelings about both Claudius and Hamlet. Claudius doesn't want to do anything to anger the people of Denmark or Gertrude, but he becomes more and more desperate to devise a plan to kill Hamlet and end the threat he poses. He tries sending him to England; then he wants England to kill Hamlet; and when that fails, he plots with Laertes to kill Hamlet in revenge for Polonius's murder.