The ghost of Old Hamlet is essential to the play as without him Hamlet would not have been presented with 'evidence' for his fathers murder. Without this the most Hamlet could level against Claudius is his dislike of the marriage to his mother, therefore he may have gone back to University and not stayed in Denmark to brood over plans to murder Claudius.
Of course, there is a big question mark over who this ghost is. It could be a goblin damned and given a familiar face in order to tempt Hamlet to his doom along with the destruction of the other characters. There is a case to argue that the ghost cannot be a good spirit because it has to go at cock crow, is giving a message of revenge and urging his son to murder. Surely if in purgatory he would not be allowed to come to Earth for that message, he should be urging his son to repentance - like the ghosts in 'Christmas Carol' by Dickens. So perhaps this is not a good ghost at all!