Why does Hamlet believe in the ghost in Shakespeare's Hamlet?
A dash of the supernatural was expected by Elizabethan audiences, who strongly believed in the influence of the other world. Also, in the Chain of Being in which the Elizabethan world found order, the king is an embodiment of God on earth, so there is a close connection between King Hamlet and the supernatural of which Hamlet would be aware.
The ghost of King Hamlet is, thus, entirely believable to Hamlet. Further, Hamlet would give credence to the ghost of his father since "there is something rotten in Denmark" and he suspects foul play in King Hamlet's death. That his father's ghost would appear to him, seeking retribution for his murder is logical to Hamlet, the Prince of Denmark.
Another condition of Hamlet's that allows for his acceptance of the ghost is his sensitive and pensive nature. Certainly, a person who is given to much existential examination allows for the influence of other circumstances of existence. Not only does Hamlet perceive the ghost and hear it speak, but he feels its presence, as well, as he proceeds through his examinations of conscience and ponderings of the value of life.