The key members of Hamlet's family are:
Hamlet's dad - the former King of Denmark who is now dead. Though he is dead, Hamlet's dad is an important figure in the play, as Hamlet thinks about him often. He also appears in the play as a ghost.
Claudius - Claudius is Hamlet's uncle, and was Hamlet's dad's brother. Claudius married Hamlet's mother (Queen Gertrude), and is therefore now the King of Denmark.
Queen Gertrude - Gertrude is Hamlet's mother and the Queen of Denmark. After Hamlet's dad died, she married Claudius, the brother to her late husband and Hamlet's uncle.