There are two notable instances in which Hamlet uses hyperbole. One occurs in the emotional scene with his mother in Act 3, Scene 4. Hamlet calls Claudius:
A murderer and a villain,
A slave that is not twentieth part the tithe
Of thy precedent lord
It would appear that he changes his mind in mid-sentence--that he intended to say that Claudius was not worth one-twentieth of her former husband--but then by adding the words "the tithe" he changes the ratio to one-twentieth of one-tenth, which would make it one-two hundredth part of her precedent lord. This is definitely hyperbole, since Hamlet has never really calculated the difference between his father and his uncle but is speaking from emotion and not from reason. Hamlet's father would have to be almost infinitely great or his uncle almost infinitesimally small to merit such a comparison.
Then there is Hamlet's confrontation with Laertes inside Ophelia's grave in Act 5, Scene 1. Hamlet tells Laertes:
I loved Ophelia. Forty thousand brothers
Could not, with all their quantity of love,
Make up my sum.
And a bit later in the same speech he says:
And if thou prate of mountains, let them throw
Millions of acres on us
Hamlet obviously tends to lapse into hyperbole when he is experiencing strong emotions. This suggests that he represses his feelings much of the time and then loses control when he releases them.
Prince Hamlet's emotional turbulence is often demonstrated in hyperbole, an unreal and obvious exaggeration, while others, like Claudius, choose to use litotes, or understatements, to describe actions or attitudes.
Hamlet employs hyperbole when he speaks to Rosencrantz and Guildenstern, telling them of his opinion of man as "a god," the greatest "beauty," and a "quintessence of dust":
What a piece of work is man...how like an apprehension, how like a god; the beauty of the world.... And yet to me, what is this quintessence of dust? (2.2.290-292)
Then in Act III, Scene 4, Hamlet speaks to his mother about his father as he shows her a picture of King Hamlet:
See, what a grace was seated on this brow?
Hyperion's curls; the front of Jove himself,
An eye like Mars, to threaten and command. (3.4.56-58)
Again, Hamlet uses hyperbole with his mythological allusions as he compares his father to the Greek gods Hyperion, Mars, and Jove.
Here is an example of litote, a form of ironic understatement:
Holding a weak supposal of our worth...
Colleaguèd with the dream of his advantage,
He hath not failed to pester us with message
Importing the surrender of those lands
Lost by his father, with all bonds of law,
To our most valiant brother....(1.2.17-23)
The use of the words "pester" and "message" are understatements. After all, Fortinbras threatens war with Denmark; he wants to take back lands that were his father's at one time. Lives can be lost as a result of this understated "pester(ing) message."