In Shakespeare's Hamlet, was the ghost a figment of Hamlet's imagination or did his father's ghost really appear to speak to Hamlet?
Shakespeare goes to a great deal of trouble in the opening scenes to establish that it was indeed Hamlet's father's ghost the men saw on the battlements. In his day Shakespeare did not have any means of making the actor look "ghost-like," such as with luminous paint, so he has him dressed in armor to make him at least look somewhat strange and unique. It was essential to establish that this actor in armor is truly a ghost and truly Hamlet 's father's spirit before Hamlet ever sees him. Shakespeare did not want to waste time when Hamlet encounters his father in having the father explain that he is...
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The play makes clear that those who see the Ghost see him/it as they remember King Hamlet as he lived. So the appearance of the Ghost is the product of individual characters' memory including Hamlet.
As for the Ghost speaking to Hamlet, the guards in Act 1 Scene 1 (and as recounted to Hamlet in 1.2 by Horatio) come to the conclusion that the Ghost wants to speak but will not speak to them. Horatio is convinced that the Ghost will speak to Hamlet. The consensus at the end of the first scene is that they will take this matter to Hamlet.
In Act 1 Scene 5 it is the Ghost that delivers the wealth of information to Hamlet. Of particular note is that the Ghost's message is a surprise to Hamlet. Also, the information is of a particularly personal nature that only a Ghost of the dead King Hamlet would know. That Hamlet later questions the veracity of the spirit, there is no question that Hamlet both saw and heard the Ghost.