Hamlet arranges to have a play performed which closely resembles the circumstances of his father's murder, to see if Claudius will react. "I have heard that guilty creatures, sitting at a play, have by the very cunning of the scene been struck so to the soul that presently they have proclaimed their malefactions."
The uncle poured poison down his father's ear. Hamlet suspects that Claudius, his uncle, is a murderer in that he suspects that Claudius killed Hamlet's father, Claudius's brother, in order to become King. When the Ghost reveals to Hamlet that he was killed by Claudius, Hamlet's immediate reaction is, "O my prophetic soul! My uncle?"--which is a pretty clear indication that Hamlet had some serious suspicions beforehand.
The Ghost. But Hamlet had already cast Claudius in this role, as can be seen in his line "O my prophetic soul! Mine uncle!". Although Claudius is certainly a murderer, and a wrongdoer, he is much more sympathetic when we do not look at him through Hamlet or his father's eyes.
Hamlet is disposed to believe ill of Claudius. He dislikes him and resents his marriage to his mother. That is why when the Ghost names Claudius as his murderer, Hamlet says "O my prophetic soul!"--he had already suspected as much. Because we hear Claudius admit his guilt (just before the "To Be or Not to Be" speech and in Claudius's "O my offence is rank" soliloquy) we know that the ghost is telling the truth. Otherwise, it might be plausible to believe that Claudius is innocent, as everyone including Gertrude seems to believe.
Hamlet was causing problems for Claudius. His antics had disturbed the court and the kingdom. His killing of Polonius had deprived Claudius of his counsellor and had put him in an awkward position, since a murderer ought to be tried and put to death yet that would be politically unwise. Finally, Hamlet was on to him. Claudius had a pretty good idea that Hamlet knew that he had killed Hamlet Sr. and how it was committed. All this meant that Claudius was worrying about Hamlet instead of running the kingdom, which he had been doing quite well before all this. Claudius says, "like the hectic in my blood he rages."--it is becoming an obsession.
Hamlet was a mass murderer. First he stabbed Polonius while he was evesdropping behind a curtain. Then he forged an order for Rosencrantz and Guildenstern to be put to death. Laertes was next murdered by Hamlet in the Fencing match where Laertes had the posioned sword. Hamlet ended up with it and cut Laertes. Finally, Hamlet killed Claudius. That's five people.
Claudius hopes that Hamlet will be killed in England. Claudius is the King and is also the brother of Hamlet.
Claudius was hamlets Uncle and later became his stepfather because his mother Queen Gertrude married Hamlets uncle "Claudius".
His brother, Claudius, came out into the garden while King Hamlet was sleeping and poured poison into his ear.
Claudius is married to Gertrude (Hamlet's mother and widow to the former king).
Polonius tells King Claudius that Hamlet's erratic behavior is the result of his love for Ophelia. He believes that Hamlet's madness stems from Ophelia's rejection and the resulting emotional turmoil. Polonius is convinced that this unrequited love is the key to understanding Hamlet's condition and intends to prove it by observing their interactions.
Gertrude, Hamlet's mother and wife of his Uncle Claudius, King of Denmark.
Hamlet's uncle Claudius killed Hamlet's father (called Hamlet Sr.). The ghost of Hamlet Sr. comes back from the dead and tells Hamlet Jr. so. Hamlet Jr. confirms this by re-enacting the murder as a play, which upsets Uncle Claudius.