The Ghost appears to Marcellus and Bernardo in Act I.i. It does not speak. It exits and re-enters (does that count as a separate appearance?) Then it exits again. The Ghost had appeared the previous two nights to these same two guards, but it was before the action of the play (does that count?)
The Ghost appears to Horatio, Marcellus, and Hamlet in Act I.iv. It beckons to Hamlet and exits with Hamlet to the next scene. The Ghost opens Act I.v, speaking to Hamlet on another platform (does that count as an appearance?) It's a new scene, but Shakespeare didn't write in the scenes in the original Folio, so I don't know. It speaks to Hamlet in scene v and tells him of his murder.
The Ghost appears to Hamlet in the famous closet interview with Gertrude and Polonius (hiding) in Act III.iv. Gertrude and Polonius cannot see the Ghost, only Hamlet. The Ghost warns Hamlet not to harm his mother.
So, the Ghost appears in 4 different scenes. It bridges two scenes, so it only really makes 3 separate appearances, unless you count the appearances before the play. If that's the case, then it appears 5 times. The Ghost has 5 entries and 5 exits.
The ghost shows up three times in the play. It first appears to Horatio, Marcellus and Bernardo. (We learn it had already appeared before to the guards, but we don't witness this happening.) After seeing the ghost, Horatio persuades Hamlet to wait for it, and it shows up again, taking Hamlet off for a private conference about avenging its death. It later appears to Hamlet alone in Act III. Gertrude thinks Hamlet is mad, because she cannot see the ghost and doesn't know who he is talking to.
The fact that the guards and Horatio also see the ghost in the first act reinforces the idea that it is real, and not simply a figment of Hamlet's imagination. Horatio, especially, tends to be a rationalist, so his sighting of the ghost and the fear it inspires in him, lends credence to its reality. The ghost apparently can only appear at night, for it disappears when the cock begins to crow.