Horatio shows he believes in ghosts in several ways. He tells Francisco and Bernardo that the ghost looks like King Hamlet, whose death has recently occurred. So it is obvious that he admits he sees something. Then, he tries to get the ghost to talk and seems genuinely frightened at its appearance. Finally, he suggests that they must find Prince Hamlet to see the ghost, in hopes that the ghost may speak to his still-living son. This begins one of the most common motifs of the play: reality or deception. The question becomes, "Is this really the ghost of King Hamlet or some evil spirit bent on destroying Denmark? "
After the ghost's appearance, Horatio exclaims, "Before my God, I might not this believe / Without the sensible and true avouch / Of mine own eyes." In other words, he had to see the apparition to believe in it.
When the ghost appears again, Horatio addresses the spirit and tries to get him to speak. The ghost, however, does not respond and exits the scene again. At this point, Horatio, Bernardo, and Marcellus decide that they must tell Hamlet what they have encountered that evening.