What are some examples of figurative language in Hamlet?

There are numerous examples of figurative language in Hamlet but here are just a few of them.
Let's start with a simile. This is a figure of speech which involves the comparison of two different things using "as" or "like." Obvious examples include "As strong as an ox," or "He eats like a pig."
In Act I Scene III of Hamlet Polonius, as is often the case, is dispensing advice, this time to his son, Laertes:

This above all: to thine own self be true,
And it must follow, as the night the day,
Thou canst not then be false to any man.

If you're true to yourself, then it will follow naturally, as night follows day, that you can't be false to anyone else.
 
Then we have personification, or attaching human qualities to things that aren't human:



When sorrows come, they come not single spies
But in battalions. (Act IV Scene V)





Claudius is lamenting the number of tragedies that poor Ophelia has recently suffered. Her sorrows are compared to battalions, large bodies of troops. Another way of saying the same thing is the old expression "It never rains but it pours." In other words, bad things tend to happen all at once.
 

Metonymy is a word or phrase used to stand for another word of phrase that's closely related. So for example, "suits" is often used to refer to business executives. In Hamlet the ghost of Hamlet's father uses metonymy to relate the particulars of his murder by Claudius:


Tis given out that, sleeping in my orchard,
A serpent stung me. So the whole ear of Denmark

Is by a forgèd process of my death
Rankly abused. (Act IV Scene V)



The "whole ear" is used in place of the whole nation of Denmark. Claudius' murder of Hamlet's father (by pouring poison into his ear) wasn't just personal; it was attack on the country as a whole.
 
Finally, we have paradox, a kind of contradiction:


I will bestow him and will answer well
The death I gave him. So, again, good night.

I must be cruel only to be kind. (Act III Scene IV)



Most people are familiar with that expression. Indeed, like so many expressions from Shakespeare it's become part of our everyday language. Hamlet is telling his mother that if he kills Claudius then he'll eventually bring some measure of peace to Gertrude. His act of cruelty will, paradoxically, lead to kindness.