How does Shakespeare use language and dramatic effect in act 4, scene 3?

Much of the drama in act 4, scene 3 stems from Hamlet's seemingly cruel cat-and-mouse game with Claudius, Rosencrantz, and Guildenstern over the whereabouts of Polonius's corpse. What's dramatic is not so much the question of what's happened to the dead body but Hamlet's reaction to events. Although he didn't intend to kill Polonius, he takes his death as a golden opportunity to keep up the facade of madness. In doing so, he resorts to extensive wordplay, adding to the macabre sense of playfulness that epitomizes the scene.
When Claudius asks Hamlet where Polonius is, he tells him that Polonius is at supper:

Not where he eats, but where he is eaten. A certain convocation of politic worms are e'en at him. Your worm is your only emperor for diet. We fat all creatures else to fat us, and we fat ourselves for maggots.

Polonius is indeed "at supper;" but he's on the menu, providing a sumptuous feast for worms. Hamlet also uses a political pun, when he says that "Your worm is your only emperor for diet." The Diet of Worms was an assembly called by the Holy Roman Emperor in 1521 to examine Martin Luther, the renegade monk who founded Protestantism.
Hamlet's callous disregard for Polonius adds to the general impression that he is not of sound mind. He soon warms to his theme, toying with Claudius like a cat with a mouse in its claws:

A man may fish with the worm that hath eat of a king, and eat of the fish that hath fed of that worm.

This line foreshadows the graveyard scene where Hamlet muses on the common fate of the highest and the lowest alike. Here, he's saying to Claudius that a humble man can catch a fish with a worm that has eaten the remains of a king:

Nothing but to show you how a king may go a progress through the guts of a beggar.

Hamlet's rather gross, repellent language has a serious point. As well as burnishing his credentials as a dangerous madman, he's also reminding Claudius of the fate that will one day come to him, as to everyone else. Hamlet is making a none too subtle threat here. Claudius doesn't need reminding that he will die one day, but the fact that Hamlet mentions this at all is disturbing for his wicked stepfather. It gives the impression that Hamlet is ready to make a move—to do to Claudius what he's just done to Polonius. This battle of wits and wills heightens the dramatic tension between the two men, forcing Claudius to make a move of his own and send Hamlet off to England with the intention of having him assassinated.