What is the role of madness in Hamlet and King Lear?

There are, of course, different ways to approach and interpret madness in these plays. One could argue that madness in both plays is a response to corruption. In Hamlet, Hamlet must cope with the possibility—and then the reality—that his uncle murdered his (Hamlet's) father to gain the throne. This realization of his uncle's capacity for evil makes Hamlet aware of the pervasive evil—described in the play as a disease or rot—permeating the entire Danish court. The world seems to Hamlet to have gone mad, and he hardly knows where to turn.
Likewise, King Lear must deal with the treachery of his close relatives, in this case his two eldest daughters, who lied to him to get his power and now are willing to cast him off and let him die. His world, too, seems to have gone mad. He can hardly believe his own children would treat him the way these two have.
Both Hamlet and Lear respond to worlds that seem to have gone mad by experiencing spells of madness themselves. Lear goes into a frenzy of anger and madness during the storm on the moor. Hamlet goes into a frenzy of anger and madness in Gertrude's chamber, during which he mistakenly kills Polonius.
It is said in psychology that the healthiest member of a dysfunctional family will often be labelled the "crazy" one by the others, because he or she calls out or names the dysfunction that everyone wants to pretend doesn't exist. Both Hamlet and Lear play this role in their respective plays. While acknowledging their own flaws, both cling to a sound moral worldview despite being surrounded by evil, and this moral compass paradoxically drives them to temporary madness when they confront the evil in their own families and courts.

I would argue that madness in Hamlet and King Lear ironically allows characters to more clearly understand truth. In Hamlet, Hamlet's decision to feign madness (although, to be sure, it's unclear whether he actually is pretending; he might very well be mad) allows the prince to discretely investigate his uncle Claudius' crimes. After all, no one would expect a crazy adolescent to be capable of snooping around. Thus, though it might take him an agonizingly lengthy time, Hamlet is able to discern the truth of his father's murder and Claudius' treachery. Similarly, in King Lear, Lear only understands the true nature of his daughters' affection (or, in the case of Regan and Goneril, the lack thereof) once he goes mad and wanders the wilderness. Not only that, but Lear's madness also enables him to come to some pretty insightful (and depressing) conclusions regarding human nature, as he posits the possibility that humans are the victims of the uncaring, meaningless, and vindictive whims of the universe. Thus, both Hamlet and Lear's bouts of madness allow the characters to experience a special epiphany.