Hamlet actually slips into insanity at certain moments in the play. Is this true, or is Hamlet merely play-acting insanity? Give evidence.-

The only evidence one might have for whether or not Hamlet
is "truly" insane in any given moment in the play is what he says or what others say
about him.  The problem with relying on what characters say about themselves or others
is that they are not always telling the truth.  How to
know?

Well, you can't know for sure, actually.  That's why
they call it a play, and it's one of the reasons that Hamlet, in
particular, has fascinated audiences for centuries.  There are as many choices to make
about whether Hamlet is insane or not (and at which moment he might or might not reveal
actual insanity or that he is merely "play-acting") as there are actors willing to
attempt the role.

I can give you evidence of what Hamlet
says both to Horatio and Marcellus in Act I and, again, in Act III to Gertrude about his
"play-acting," but this is not some sort of conclusive evidence of what is really going
on with Hamlet.  It is pretty widely considered to be true that a person who is insane
is usually the last one to know.  Most insane people don't recognize themselves as
such.  Here are the scenes and text from Acts I and III in which Hamlet reveals that his
insanity will be and is a ruse:

readability="14">

Here as before, never, so help you
mercy,

How strange or odd some-er I bear myself
--

As I perchance hereafter shall think
meet

To put an antic
disposition on --

(I,v,
177-79)

and, in III, iv, in
attempting to convince Gertrude not to allow Claudius back into her
bed:

readability="21">

Queen

What
shall I
do?

Hamlet

Not
this, by no means, that I bid you do;

Let the boat King
tempt you again to bed,

. . .And let him, for a pair of
reechy kisses,

. . .Make you to ravel all this matter
out

That I essentially am not in
madness,

But mad in
craft. (lines 182-
190)

There are numerous
moments in the play in which the other characters (in fact, nearly all the characters
except Horatio) discuss Hamlet's "madness" as if this state of mind is a given fact. 
Polonius and Claudius go farther, attempting to discover the reason for his "madness." 
However, other characters' opinions of his behaviour don't solve the question of whether
he is play-acting or not.  These characters are only witnessing his actions, which is
not any sort of evidence for what is going on internally with
Hamlet.

Perhaps the largest body of evidence that can be
put forward for Hamlet's sanity are his lucid and eloquent soliloquies -- his
conversations with the audience, during which he reveals his inmost thoughts.  These do
not seem to be the ravings of a mad man, but rather the tormented thoughts of a lucid,
highly intelligent, grief-stricken man.

All of this being
said, this is a question that has been debated through the centuries.  For more on
Hamlet and madness versus play-acting, please follow the links
below.