Why does Hamlet get upset with the gravedigger?

Hamlet is upset with the gravedigger because the gravedigger does not seem to treat his task with the proper gravity. Hamlet asks Horatio, of the gravedigger, "Has this fellow no feeling of his business? He sings in grave-making" (5.1.67-68). Having recently lost his father, Hamlet is especially aware of the real-life tragedy of dealing with the death of someone one loves. When the gravedigger digs up a skull, Hamlet says, "That skull had a tongue in it and could sing once" (5.1.77-78). He considers the lives of the people whose bones the gravedigger digs up, thinking about what they might have been like during their lives, humanizing them in a way that the gravedigger does not. He becomes a bit philosophical as a result.
Once Hamlet begins to converse with the gravedigger, he becomes a little irritated at how literal the man is in his speech. The gravedigger says that the grave is not for a man, and is not for a woman, but it is for a person who used to be a woman during her life. "How absolute the knave is! We must speak by the card, or equivocation will undo us" Hamlet exclaims (5.1.140). The gravedigger's speech is so literal that it is difficult to actually have a conversation with him.

Hamlet's upset with the gravediggers because they're not treating the dead with the respect that they deserve. The gravediggers are comic characters, a couple of clowns who spend most of their time joking, singing, and engaging in witty word-play. Hardly the attitude we might expect from someone entrusted with such a solemn task. The gravediggers have become so used to being in close proximity to the dead that they've almost become immured to the sordid details of their work. Horatio understand this:

Custom hath made it in him a property of easiness. (act 5, scene 1)

But Hamlet is less understanding. That skull that one of the gravediggers is throwing about while he cheerfully sings once belonged to someone. It once had a tongue inside it; the person it belonged to also sang once upon a time. Nevertheless, Hamlet soon catches the infectious spirit of the gravediggers' gallows humor and begins to make jokes at the expense of the dead, musing aloud on whether one of the skulls once belonged to a greedy, grasping lawyer.