What are some examples of when Hamlet acts insane?

Hamlet is experiencing some personal conflict as a result of the information he gets from his father (the ghost) that his uncle is a murderer.  His father wants to be avenged, and Hamlet wants to comply, but Hamlet is a complicated soul.  Just killing his uncle is not his style.  His mother married his uncle, a fact that makes him very angry.  He's conflicted about killing his uncle.

While Hamlet is trying to figure things...

Hamlet is experiencing some personal conflict as a result of the information he gets from his father (the ghost) that his uncle is a murderer.  His father wants to be avenged, and Hamlet wants to comply, but Hamlet is a complicated soul.  Just killing his uncle is not his style.  His mother married his uncle, a fact that makes him very angry.  He's conflicted about killing his uncle.

While Hamlet is trying to figure things out, he behaves in a very strange manner the many people would consider crazy.  To some people, the first sign of his insanity is seeing and speaking to ghosts.  Even if you accept that, there are other instances of Hamlet behaving irrationally and rudely.

One example is Hamlet’s behavior toward Ophelia.  By all accounts Ophelia is a sweet girl, and Hamlet was sweet on her.  Whatever the extent of the relationship, Hamlet’s treatment of Ophelia is beyond harsh.  It is uncalled for.

HAMLET

You should not have believed me; for virtue cannot
so inoculate our old stock but we shall relish of
it: I loved you not.

OPHELIA

I was the more deceived.

HAMLET

Get thee to a nunnery: why wouldst thou be a
breeder of sinners? I am myself indifferent honest;
but yet I could accuse me of such things that it
were better my mother had not borne me … (Act 3, Scene 1)

Hamlet’s harsh and crude treatment of Ophelia is nothing compared to what he does to her father.  He kills him, more or less accidentally, and then when his uncle tries to get Hamlet to tell them where the body is, Hamlet uses it as another opportunity to prove that he is crazy. 

KING CLAUDIUS

Now, Hamlet, where's Polonius?

HAMLET

At supper.

KING CLAUDIUS

At supper! where?

HAMLET

Not where he eats, but where he is eaten … (Act 4, Scene 3) 

It’s funny in a completely morbid way.  Hamlet doesn’t want anyone to guess his intentions regarding his uncle, so he wants to make everyone think he is crazy.  He tells his friends Rosencrantz and Guidenstern that he knows " a hawk from a handsaw," and he is wily enough to avoid his uncle's plot to murder him.