Who is the ghost and why does it tell hamlet it must walk the night and burn in hell during the day?

Answer 1

The ghost is Hamlet's father. It tells Hamlet that it must walk the night and burn in Hell in the day perhaps to evoke a feeling of pathos in Hamlet. The reason the ghost would be subject to this torture is due to it being stuck in Purgatory.

Answer 2

The ghost is Hamlet's "father's spirit" as he describes himself. He is not actually in Hell when he is not haunting Elsinore castle, but rather in purgatory, a place where "the foul crimes done in my days of nature are burnt and purged away". Eventually he is going to heaven once he has atoned for his sins. He tells Hamlet this (despite saying that he is "forbid to tell the secrets of my prison house") in order to get Hamlet angry at the guy that murdered him, Hamlet's Uncle Claudius. He wants Hamlet to revenge the murder by killing Claudius, which is why he has been hanging around the battlements of the castle.