The only ghost that appears in the play is Hamlet's father's ghost. Hamlet's father was the king prior to the start of the play. The dead king is sometimes referred to as Old Hamlet. Claudius is the king's brother and Hamlet's uncle. Claudius kills his brother and becomes king.
The slain king's ghost comes back to tell Hamlet that Claudius is the murderer. Hamlet swears to avenge his father by killing Claudius. This ghost first appears in Act 1, Scene 1. Hamlet first sees the ghost in Scene 4 and speaks with it in Scene 5.
Claudius dies in the final scene of the play, along with many other characters. But there is no evidence of his ghost appearing. However, there may be versions of Hamlet in which the director takes an artistic license and uses the notion of Claudius's ghost in some way. I do not know of this occurring but it is possible. In the notes to the soundtrack of Kenneth Branagh's film version of Hamlet, there is a note about Claudius's ghost incarnating via a statue of himself early in the play. If this does occur in the film, it is to establish the evil spirit of Claudius and to create an atmosphere of the supernatural. But in the original version of the play there is no ghost of Claudius.