Claudius is an evil character in that he plots to kill his brother and marry his sister-in-law to secure his spot on the throne. Later in the play, when he begins to suspect Hamlet, he plots to kill him as well by sending him to England with a letter which requests his immediate execution. Claudius is also behind fueling Laertes' anger and desire for revenge on Hamlet over his father's untimely death.
A case can be made that Hamlet is a virtuous young man at the beginning of the play, his sense of morality deeply rooted in his religious beliefs, but that he changes dramatically in the play and succumbs to evil himself by its conclusion. Hamlet's actions in avenging his father's death are perfectly justified within the context of Hamlet as a revenge play, but when examined within the greater morality that had framed Hamlet's life and that is referenced often in the play, his actions seem consistent with the evil of those that surround him.
Hamlet is truly a dynamic character. Upon first returning to court, the idea of committing murder under any circumstances shakes him profoundly. He fears for his immortal soul; even the idea of taking his own life fills him with fear of God's judgment. However, after Hamlet kills Polonius while Hamlet is in a state of unbearable emotional suffering, Hamlet immediately spirals downward from his previous standards of moral behavior. He plots and accomplishes the deaths of Rosencrantz and Gildenstern with cold calculation; his actions are deliberate and absent any moral inhibition or sense of guilt. This marks a profound change in Hamlet, suggesting that even he, as morally sound as he had once been, is also capable of evil acts.
There is clearly evil in the entire court of Denmark. They are the "something rotten." Polonius is a hypocritical fool, and while both hypocrisy and foolishness should be punishable, hypocrisy as deceit is certainly evil. King Claudius covets both his brother's throne and his wife. Aren't these actions against the 10 commandments? (Don't forget that Denmark is Catholic at the time of the play.) And, while Hamlet is not evil, he does sin some as murders Polonius and he is deceitful to Ophelia and manipulates her, actions which may lead to the innocent girl's suicide. However, for the most part he is a noble, albeit tragic, man.
If sin is evil, then Hamlet is guilty of being evil when he kills Polonius. Despite the fact that Claudius (which, of course, is who Hamlet thought he was killing) was a murderer, Hamlet had no right to take his life. It is not man's place to seek revenge, and doing so is in direct defiance of God's law. Sin.
I disagree with #2. I think we can infer that Claudius was evil in his act of killing King Hamlet. He clearly had designs on both his brother's crown and his brother's wife, and waited until he saw his opportunity and then seized it. Although he later repents and confesses and shows remorse, I think we can clearly count his actions as being evil in terms of his calculated intentions.
The only possible actual evil I see in this story is in Claudius. He is the only one who does a really bad thing due to (as far as we know) greed or the desire for power. He seems to have killed his brother simply to have taken the throne.
As for the characters you mention, I agree with the first post. I do not think that they are in any way evil.
If your response is limited to only an analysis of the three characters mentioned in your posting, then I would say no, there is no evil in these characters. They each have there flaws, but their ultimate intentions are not malicious, premeditated or evil.
We know very little about the life and actions of King Hamlet, but he seems to have been a good king and a successful warrior. He lawfully engaged in a battle with King Fortinbras, won, and gained land as a result. This is not evil, this is just a matter of life in these times. It is what kings did in order to secure their kingdoms. His return as a ghost isn't evil. He needs to have Prince Hamlet know that Claudius murdered him and ask for vengeance. Hamlet feels a need to assure himself that the ghost is a "true" ghost and not a devil in disguise, but once that is done he is free to act. Seeking vengeance isn't considered evil -- it is an act of justice.
Polonius isn't evil; he is a conniving, sneaky spy, but he isn't trying to cause permanent damage to people. He wants to protect his and his family's reputation and prove himself important and valuable to the king. His spying ways result in his accidental death by Hamlet.
Hamlet himself is not evil. He is actually extremely cautious about committing acts that would be considered evil. He verifies the story of the ghost by getting proof that Claudius did, in fact, kill King Hamlet. He doesn't kill Claudius when it appears he is at prayer. He accidently kills Polonius thinking it is Claudius who is hiding behind the curtains. He rewrites the letter to England as a measure to save his own life, not to unjustly have Rosencrantz and Guildenstern killed. He stabs Laertes with the poisoned sword only after he was struck by it first, and even at that moment he doesn't realize the sword was poisoned. He does kill Claudius, but that is the act of vengaence that he has been struggling to complete through the whole play.
Hamlet is rather ambiguous in his actions. To some readers, the executions of Rosencrantz and Guildernstern may have been unnecessary, whereas to other it would seem critical to his heroic plight. Regardless of either interpretation, Hamlet has blood on his hands, the blood of individuals who held him in a high esteem.
Additionally, the argument can be postulated that Hamlet's rejection of Ophelia contribute to her despair, insanity, and ultimate suicide. Although he was pretending to be mad during this scene (or perhaps he wasn't, an entirely different argument) he spurned her. Also, he may have indirectly contributed to her emotional collapse by killing her father. The text proves that this was indeed a mistake, but is someone who decides to stab someone, without confirming that they are indeed the intended target, really all that magnanimous?