Claudius is the king of Denmark, brother of the dead king, married to the widow Gertrude and therefore uncle and stepfather to Hamlet. He is the one the ghost accuses: "The serpent that stung your father's life now wears his crown." The official story was that the king had been bitten by a serpent while sleeping in the garden.
In Act I, scene 2, Claudius announces his marriage to Gertrude, sends letters to the king of Norway to contain Fortinbras, and asks Hamlet to stay at Elsinore and think of him "as a father."
At The Mousetrap, seeing the poisoning in the orchard staged, he rises and asks: "Give me some light. Away!" Alone, in Act III, scene 3, he confesses "A brother's murder!" and says he cannot pray because he is still in possession of the "effects for which I did the murder": "My crown, mine own ambition, and my queen."
After Polonius's death, he says he cannot punish Hamlet by law because the prince is loved by the distracted multitude, and he sends him to England with letters that demand the prince's death. When Hamlet returns, he arranges with Laertes a duel with a poisoned sword and prepares a poisoned wine cup as well. In the duel, the cup kills Gertrude, and, accused by Laertes, Claudius is pierced by Hamlet's blade and forced to drink the wine.