In Shakespeare's Hamlet, the supernatural is a major element of the story.
Shakespeare is writing to an Elizabethan audience that has been born into the belief of the power of the supernatural, and fed a diet of tales of what the "other-worldly" are capable of doing.
...Elizabethans truly believe in the existence of these spirits...they even have somewhat of a list as to the characteristics of ghosts. These characteristics are embedded within Shakespeare’s writings and are followed to exactness. The first and foremost characteristic is that ghosts are considered evil spirits that impersonate the deceased. This characteristic helps to provide a plot such as in Hamlet where when the father’s ghost first appears, Hamlet does not know whether he is good or evil. Therefore, when the ghost seeks out Hamlet to do his bidding, Hamlet does not know if he should follow the apparition’s bidding.
I think that seeing the ghost goes a long way to convincing Hamlet that the ghost is an "honest" one at that moment. This may be why Hamlet initially tells Horatio that he believes the apparition. However, when the night is over and daylight returns, so does reason. Hamlet (a personification of a character with Elizabethan viewpoints) will think twice about what he saw the evening before. Being concerned over the Ghost's "credibility" would be the result of Hamlet's common sense challenging what Hamlet had witnessed under a cover of mystery and darkness the night before.
Hamlet, especially, provides the common perception of a ghost. This apparition is seen as a tormented figure who suffers purgatory because of his murder and wife’s adultery. And to reiterate the common beliefs, Hamlet is torn between thinking of the ghost as his father in life and experiencing and seeing the actual disturbing apparition. When ghosts are not actually seen...when it involves the imagination of the character, the audience still perceives the effects of the apparition as it seeks to persuade a human to enact revenge. This is due to the widely accepted belief that a ghost does not have its own powers but must use the power of persuasion to exact revenge. Shakespeare uses this perception to develop a sense of doubt and eventually forces a revelation upon the audience. This pagan element is accepted in its entirety by the audience as a true possibility. The use of ghosts within Shakespeare’s works is both entertaining and believable since its actual conception is derived from society’s beliefs. (www.writing.com)
Because the Elizabethan audience believed unequivocally in the power of ghosts to appear after death, there would have been no doubt that Hamlet indeed saw a ghost. Old Hamlet's ghost did not have the power to make Hamlet exact revenge: Hamlet had to agree of his own free will. However, the audience would still harbor doubts as to the identity of the ghost: is it, in fact, Hamlet's father? Elizabethans believed a ghost could be evil, pretending to be a lost loved one, thereby tricking a living person to do its bidding (most of the time exacting revenge).
Because the Prince has so much to lose if he aligns himself with an evil spirit, he must be certain that he is dealing with his father's ghost. This is why he hesitates so long in acting against Claudius, even though his heart tells him it's true; once Claudius reveals his guilt at the play, Hamlet has the proof he needs.