In Shakespeare's Hamlet, what is the cause of Hamlet's madness? Use details from the play to support your answer.

The issue of Hamlet's madness in Shakespeare's Hamlet is complex to say the least.  The more central issue is whether Hamlet is "mad" or is just faking it.  But if one assumes he's mad and just wants to know what the cause of that madness is, the issues are a little clearer.  First and most importantly, if Hamlet is mad it is due to grief over the death of his father/king.  For example, Claudius and Gertrude are aware of his depression early in Act I and exhort him to recover from the loss.  "How is it that the clouds still hang on you?" (I.ii) asks Claudius.  Gertrude follows:

Good Hamlet, cast thy nighted color off,

And let thine eye look like a friend on Denmark.

Do not for ever with thy veiled lids

Seek for they noble father in the dust. (I.ii)

Soon after, still in scene two, Hamlet gives his famous "too too sallied flesh" soliloquy, and his depression becomes even more evident.  He, in fact, wishes the "Everlasting had not fixed/His canon 'gainst self-slaughter."  Hamlet is suicidal due to his father's death.  One could cite numerous other examples of Hamlet's madness being due to grief over his loss.

Other issues that one could study are also present in the play:  Polonious suggests Hamlet's madness results from Ophelia's rejection of him and Hamlet certainly suffers due to his perceived inaction, just to name two.