When the Ghost appears in Act I scene 1, it is clear that he resembles the Old King Hamlet, as Horatio says that it looks "Most like" the original king, even down to the armour that he wore. Note what Horatio says about the armour that the Ghost is clothed in:
Such was the very armour he had on
When he th'ambiotious Norway combated.
Not only is he wearing exactly the same armour that he possessed whilst alive, but even his facial expression is the same as the kind of facial expression he used to wear when alive. In response to promptings from Marcellus, Horatio is forced to concede that the Ghost resembles the Old King Hamlet in every way, and therefore, if appearances are to be believed, is the shade of this august personage who died before the beginning of the play.