Was Hamlet actually going mad in the play? Or was it just an act?

Determining whether or not Hamlet was actually "mad" has been a favorite topic of Shakespeare scholars for decades. It seems to have been left open for interpretation entirely. Generally, there are two schools of thought on the topic: A) That Hamlet is MAD, and B) That Hamlet is not mad, but in his crafty way has developed a disguise of 'madness' in order to facilitate his revenge plot against Claudius. The second lends itself to a far more interesting and nuanced interpretation, and seems to be supported by Hamlet's own words. 

It is the character Polonius who first accuses Hamlet of madness, when he goes to talk to Gertrude about her son. This is in Act 2, scene 2: 

I will be brief. Your noble son is mad. 
Mad call I it; for, to define true madness, 
What is't but to be nothing else but mad? 

His words, however, must be taken with a grain of salt, as he is only trying to look out for the best interests of his daughter Ophelia. In an aside, later in the scene, however, even he concedes to the audience that there is "method" to Hamlet's madness--suggesting that it is more planned than the Prince wants to let on. Then, in act 3 scene 1, Hamlet's friend Guildenstern makes a similar remark when he describes Hamlet's madness as "crafty".

Hamlet's uncle Claudius doesn't even really believe in the truth of Hamlets "madness": 

Love? his affections do not that way tend; 
Nor what he spake, though it lack'd form a little, 
Was not like madness. There's something in his soul 
O'er which his melancholy sits on brood; 
And I do doubt the hatch and the disclose 
Will be some danger... (III, I)

But the words which should best be believed are likely Hamlet's own, in a conversation with Gertrude:

My pulse as yours doth temperately keep time 
And makes as healthful music. It is not madness
That I have utt'red. Bring me to the test, 
And I the matter will reword; which madness
Would gambol from. 

[...]

Make you to ravel all this matter out, 
That I essentially am not in madness, 
But mad in craft. (III, IV)

It seems clear then from this that Hamlet is not really "MAD", but just pretending in order to successfully carry out his plans.