Act 5.1: "Has this fellow no feeling of his business,...grave-making?" Discuss the significance of this extract in Hamlet.
The gravedigger scene in Shakespeare's Hamlet serves multiple purposes in the play.
First, it serves as comic relief, being sandwiched between the intense scenes that reveal that Ophelia has drowned, and her burial, at which Hamlet discovers that Ophelia is dead and declares his love for her and challenges Laertes. Dramatic intensity cannot be maintained indefinitely. Shakespeare lowers it with the gravedigger scene so that he can raise it again in the funeral scene.
The particular lines you quote:
Has this fellow no feeling of his business, that 'a sings in grave-making? (Act 5.1.59-60)
contribute to the theme of death, contribute to imagery of death and decay, and establish that the gravedigger is so well acquainted with death that he nonchalantly goes about his business.
Hamlet is obsessed with death throughout the play, contemplating it in his soliloquies and quipping about it every chance he gets. These words and this scene give concreteness to Hamlet's thoughts and words. The skull the clown throws out of the grave turns out to be the skull of Elsinore's court jester when Hamlet was a child. When Hamlet discovers this, he ruminates on his experiences with Yorick and creates poignancy (a quality of specialness) as he thinks of his history with the jester. Death, in effect, is personified or at least elaborated on in the form of Yorick.
Finally, the gravedigger serves as a foil to Hamlet when he sings. He reveals, when he sings as he digs a grave, a different way of handling death. He copes by singing. Whatever else Hamlet does, he does not cope with tragedy by singing.