In Shakespeare's Hamlet, where do we see that Hamlet has doubts about the Ghost?
In Shakespeare's Hamlet, there could easily be confusion about whether Hamlet has doubts regarding the specter that appears to him at the beginning of the play. In Act One, when he confronts the Ghost, he reports to Horatio and the others that it is an "honest" ghost.
Touching this vision here,
It is an honest ghost, that let me tell you. (I.v.151-152)
However, he still has his doubts. Scholars—for many years—have accused Hamlet of indecision: he is defined as a tragic hero, and as such, he has a tragic flaw (a mistake he makes that brings about his own destruction), which is said to be his inability to take action. And it is true that had Hamlet killed Claudius earlier in the play, he could have avoided the decimation of his mother and Laertes—and perhaps even Ophelia and Polonius—before the play's end.
However, there are several points to be considered. Regicide was considered a mortal sin in Shakespeare's time. The Elizabethan audiences understood this idea, and struggled with it as Shakespeare's Hamlet does: what if a king is a tyrant? Is it still a sin to kill him? So Hamlet hesitates for fear of losing his immortal soul.
Elizabethans (and Shakespeare was one of these) also believed in the supernatural—as much as they believed in anything else. They were certain that witches, fairies and ghosts existed. And while they believed a ghost could not make someone do something, they were sure it could influence a person to commit a terrible sin—thus losing his/her immortal soul in the bargain—if the ghost was not a good ("honest") ghost.
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.
And because Hamlet cannot be sure of the Ghost, he goes about trying to find evidence to prove what the Ghost has said: that Claudius murdered Old Hamlet while he was napping in the orchard—which he did at the same time every day. Hamlet acts like he is crazy to give him freedom to look for this proof—which confuses everyone—even Claudius is suspicious enough to worry that this insanity is a ploy and that Hamlet is really a threat to Claudius' place on the throne.
When the traveling players (actors) arrive, Hamlet modifies their play to include something very similar to the murder of his father. When the play is put on, Hamlet plans to watch Claudius' face and his reactions. If he is upset, this will prove the King's duplicity and guilt in stealing the throne from his brother (by murdering him).
Sure enough, when the play is presented and one actor "poisons" the Player King, Claudius stops the performance and rushes from the room. This, then, is Hamlet's proof. His is overjoyed and tells Horatio:
O good Horatio, I'll take the ghost's word for a
Very well, my lord. (III.ii.275-278)
The fact that Hamlet needed this proof supports the idea that he had his doubts about what the Ghost first told him. Now he further asserts his certain belief in the Ghost's words after seeing the King's obvious guilt.