In the biblical story, Cain and Abel are the sons of Adam and Eve, and Cain kills Abel because he is jealous of him -- he seems more favored by God. This is usually considered the first murder and this is where the connection is made. Just as Cain killed his brother Abel for shallow and selfish reasons, so does Claudius kill his brother, the King Hamlet. Once he commits the murder, he then is able to seize the things that he envied of the brother -- most noteably the thone, and next most important, his sister-in-law Gertrude whom he proceeds to convince to marry him within a month of the King's death. Claudius admidts in Act 3 that he did in fact kill Hamlet and then questions how he could ever be forgiven, especially considering that he is not willing to give up the things he gained from it, namely "my crown, my own ambition, and my queen." He was jealous of his bother's life, and killed him in order to take the "life" for himself.