I don't see how an audience can find anything redeemable in Claudius's character which would cause them to think Claudius was unjustly punished. He is single-minded and driven throughout the entire play to hold on the throne at all costs. He steadily intensifies his quest to destroy Hamlet and even lets Gertrude die for that cause. His one moment of guilty contemplation in Act 3 ultimately reveals that he is incapable of real guilt. He asks point-blank "what form of prayer can serve my turn?" He knows there are no prayers for people who don't want to give up what they have gained from their sins. It is an extra special irony that he is ultimately destroyed by the plot of poison cup and sword that he came up with in the first place.
Claudius definitely killed his brother; Claudius definitely killed a king; and Claudius probably stole (seduced?) his brother's wife. Before he dies, Claudius has to watch Gertrude (a woman he seems to love) die and know that he caused her death. It sure seems as if what happened to him is little enough to pay for the damage he caused out of his own lust for a throne and a woman. He probably deserved more earthly punishment; however, according to Elizabethan theology, his damnation is eternal.
Claudius killed his brother. He deserves to die. Also, he spends much of his time thereafter trying to kill Hamlet. He deserves to die. The poison that he intended for Hamlet serves him right. Justice prevailed. Sadly enough, Hamlet has to die as well. There is no question about it, Claudius deserves his punishment. It just did not happen soon enough for the reader.
King Claudius certainly deserves the punishment for regicide. That he is guilty of having killed King Hamlet is evinced in Act III, Scene 3 with the soliloquy of 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,
And, like a man to double business bound,
I stand in pause where I shall first begin, (3.3.38-44)
In effect, Claudius is the one who has set in motion all the things that are "rotten in Denmark": He has negotiated the nefarious plot with the former friends of Hamlet to engage the English king in Hamlet's murder, he is the cause of Gertrude's anguish over having betrayed Hamlet and his father; moreover, he brings about her untimely death when she drinks from the poisoned cup. Claudius is clearly the cause of Hamlet's torturous melancholy and self-debate and suspicion of his mother, Ophelia, and his friends. And, his intervention in Laertes's plans to duel Hamlet by poisoning the tips and have poison in Hamlet's is certainly full of evil intent.
At the end, Horatio seems to indicate that Claudius has under the law of retribution
So shall you hear
Of carnal, bloody and unnatural acts;
Of accidental judgments, casual slaughters;
Of deaths put on by cunning and forced cause;
And, in this upshot, purposes mistook
Fall'n on the inventors' heads. All this can I
Truly deliver. (5.2.395-401)
Fortinbras, too, agrees, praising Hamlet "To have proved most loyal" in all that has gone "amiss."
I don't think you can feel too bad for Claudius. He is guilty of a heinous crime and he knows it. We can see this in the fact that he is not even able to pray. Because he has done such a terrible thing, we can certainly say he deserves to be punished. Death is not too harsh of a punishment for him, particularly in view of the time in which this play was written.
Absolutely! He spent all of his time and effort plotting against Hamlet, especially as it becomes clearer that Hamlet is "on to" Claudius.
He won't do the dirty work himself, however. He had Polonius spy on Hamlet, which put an undue amount of grief on Ophelia. Then, he enlisted the aid of Gertrude, which led to Polonius' death. When Laertes came to avenge his father, Claudius used him to destroy Hamlet, and he was killed for his efforts. Claudius' ambition was at the root of all this.