There is little doubt that Hamlet's madness is indeed feigned. He explicitly mentions this in act 1, scene 5 when he refers to putting "an antic disposition on." But there are further instances where Hamlet makes it clear what he's up to. In act 2, scene 2, Hamlet is conversing with his old school chum, Guildenstern:

"I am but mad north-north-west: when the wind is
southerly I know a hawk from a handsaw."

In other words, Hamlet is only mad when it suits him. Otherwise, he's as sane as anyone else. But there's really no need for Hamlet to keep confessing to his sly trickery. Just about everyone in the play comes to realize, sooner or later, that his alleged madness is all just an act:

"Though this be madness, yet there is method in't."

That's Polonius, earlier on in act 2, scene 2. Claudius isn't wholly convinced, either, even though it's in his interests to think that Hamlet really has taken leave of his senses:

"What he spake, though it lack'd form a little. 
Was not like madness." (act 3 scene 1)

Hamlet's feigned madness admirably serves his purposes. But it also serves the internal dynamic of the play by conveying real meaning to an audience. If Hamlet really were insane, then it would be difficult, both for us and the play's sane characters, to have any idea of what's going on. And with that, the play would lose psychological complexity and dramatic depth.

In Act 1, Scene 4, Hamlet, accompanied by Horatio and Marcellus, encounter the ghost of Hamlet's father, the previous king of Denmark. This is the same ghost that Horatio, Marcellus, and Bernardo saw in the first scene of the play.

In Act 1, Scene 4, the ghost beckons Hamlet to follow him and Hamlet agrees, despite the warnings of his friends.

Act 1, Scene 5 begins with a conversation between the ghost and Hamlet. The spirit of Hamlet's father tells him how he was killed and urges Hamlet to take revenge against Hamlet's uncle Claudius. Hamlet agrees to his father's wishes and the ghost disappears.

After this, Marcellus and Horatio appear and talk to Hamlet. During the course of their conversation, he tells them of his plan to pretend to be insane, a tactic that will help him to carry his revenge.

Here, as before, never, so help you mercy,
How strange or odd soe'er I bear myself,
As I perchance hereafter shall think meet
To put an antic disposition on,
That you, at such times seeing me, never shall,
With arms encumber'd thus, or this headshake,
Or by pronouncing of some doubtful phrase,
As 'Well, well, we know,' or 'We could, an if we would,'
Or 'If we list to speak,' or 'There be, an if they might,'
Or such ambiguous giving out, to note
That you know aught of me: this not to do,
So grace and mercy at your most need help you, Swear.