First, as a quick correction, Cladius was not the one in posession of madness or insanity. The insanity and madness was an issue Hamlet himself dealt with. Claudius is the corrupt new King. He killed his brother, King Hamlet, and stole his kingdom, his wife, and his wealth. One could surmise that these actions are not done with a clear mind, but as for madness as a theme in the play, Hamlet is the character who is presented as mad.
Next, the primary reason for Hamlet's madness is the knowledge of the method of his father's death.
I am thy father's spirit,
Doom'd for a certain term to walk the night,
And for the day confined to fast in fires,
Till the foul crimes done in my days of nature
Are burnt and purged away. But that I am forbid
To tell the secrets of my prison-house,
I could a tale unfold whose lightest word
Would harrow up thy soul, freeze thy young blood,
Make thy two eyes, like stars, start from their spheres,
Thy knotted and combined locks to part
And each particular hair to stand on end,
Like quills upon the fretful porpentine:
But this eternal blazon must not be
To ears of flesh and blood.
In this speech from the Ghost of King Hamlet, he informs Hamlet about the reason for his death. He tells his son that his brother, Claudius, (Hamlet's uncle) killed him while he slept in his garden. Claudius poisoned King Hamlet in order to take the throne, the queen, and the wealth.
"Ay, thou poor Ghost, while memory holds a seat
In this distracted globe. Remember thee!
Yea, from the table of my memory
I'll wipe away all trivial fond records"
When Hamlet learns this information from the Ghost, he vows to seek revenge on Claudius for this action. His desire for revenge is the source of his madness and Hamlet continues to spiral further and further into this madness until the tragic ending of the play.