One thing we must consider is that we are not told of the reason why Claudius murders King Hamlet. Readers might assume it's because he wants the throne and the queen, but what if it's something different? We're not told, so there might be other things to consider. There is no doubt that Claudius loved his country. Perhaps Hamlet was a weak king. Is it possible that Claudius committed the murder for the greater good of Denmark. When he enumerates the passions of his life it's his country first, then the crown, then the queen. Perhaps he married Gertrude to protect her since he destroyed her protector.
Well, Gertrude must have found him sexy, eh? ; ) (At least someone did.) Ha!
In Act I, scene ii, Claudius does beseech Hamlet to remain and not return to school in Wittenberg. It would seem that he does this on behalf of the queen, for after he says,
Here in the cheer and comfort of our eye,/ Our chiefest courtier, cousin, and our son (II,ii, 116-117),
the queen repeats this request:
I pray thee stay with us...(l.119).
Whether Claudius is sincere in his request is up for debate since he may simply be acting for the sake of pleasing his new wife.
Later, in Act IV, after Rosencrantz and Guildenstern depart to find the body of Polonius, Claudius appears to have a conscience when, now alone, he reflects,
My soul is full of discord and dismay (IV, i,45.)
Of course, he still continues with his dastardly plans. Then, in Act V he tries to keep his wife, Hamlet's mother, from drinking the poison intended for her son:
Gertrude, do not drink (V,ii, 267).
But, after Gertrude drinks, he simply says in an aside,
"It is the poisoned cup; it is too late (V,ii, 269).
Only these three situations may be interpretied as showing a modicum of humanity on Claudius's part.
If he does, they are pretty hard to see. He murders his brother and then marries his wife, the widow that he has created. He arranges for the murder of Hamlet by R/G, although it never actually happens. At the end of the play hs arranges for the poison that ironically winds up killing Gertrude, the woman he murdered his brother for.
I'm going to vote for "damned smiling villain" and let others try to find some good in this scoundrel.
I really can't be as gracious as mwestwood. Claudius kills his brother, then marries his sister in law within a month of the murder. When Hamlet is understandably upset, he tells Hamlet that his grief is "unmanly" and "a fault to heaven." In other words, although it's been less than two months since your father suddenly died, snap out of it and, "think of me [Claudius] as a father. When he asks Hamlet to stay in Denmark, I think it's because he wants to keep an eye on him so that Hamlet cannot threaten his throne once outside of Elsinore. As for the death of Polonius, I think for a moment he was sad, but he buries Polonius quickly and without honor so no one will suspect anything--like what was Polonius doing in his wife's bedroom? When he orders Ophelia buried on hallowed ground after a suicide, once again I think his motives are more to prevent questions that might prove embarrassing. Finally, when his wife is about to drink poison, and he allows it, he could have stopped her by knocking the cup from her hands or telling her the truth. But by this time, Claudius is too wrapped up in protecting himself and his throne to think of anyone but himself.