How is madness portrayed in Macbeth?

After Macbeth and Banquo’s encounter with the Weird Sisters in act 1, scene 3, Banquo wonders if the two have “eaten on the insane root / That takes the reason prisoner?” (1.3.87–88). The experience is so strange that he thinks it is possible that he and Macbeth have accidentally ingested a plant that causes insanity. Macbeth, though, does not even really consider this possibility, and Banquo lets it drop immediately afterward. This interaction with the Weird Sisters, however, is what initiates Macbeth’s moral decline and descent into a kind of madness that gets worse and worse after he kills Duncan.
Just before he murders the king, Macbeth actually does hallucinate a dagger, the very weapon with which he plans to kill Duncan. It is, at first, clean, but when he looks away and back again, it is covered with gore. It is, he recognizes, a “false creation / Proceeding from his heat-oppressed brain” (2.1.50–51). He is, perhaps, already being driven mad by the guilt he feels about his plan to kill Duncan; after all, he tried to back out of the plan once, and his wife insulted and berated him until he recommitted. He doesn’t want to commit the murder, and it seems already to weigh heavily on his conscience, perhaps driving him toward madness.
This is almost certainly what Lady Macbeth fears when, after the murder, she tells him, “These deeds must not be thought / After these ways; so, it will make us mad” (2.2.45–46). Macbeth is obsessing over the fact that he could not utter the holy word Amen after he killed the king, and he is dwelling on his guilt to a dangerous degree. Lady Macbeth seems to fear that if he is unable to move on, to carry on as normal, after the murder, he will go insane. He will seem to struggle with his sanity later, and ironically, so will she, despite her apparent remorselessness now.
Later, after Macbeth has arranged for the murder of Banquo, but prior to the actual murder taking place, he tells Lady Macbeth, “O, full of scorpions is my mind, dear wife!” (3.2.41). This certainly sounds like a description of an overwrought brain. He says that it would actually be better to be “with the dead,” who are now at peace, than to be living and “on the torture of the mind to lie / In restless ecstasy” (3.2.24–25). Macbeth is having trouble sleeping, which is driving him to distraction—it feels like torture—and he comments that his failure to achieve any peace at all is so terrible that he would rather be dead than experience it.
After the disastrous state dinner, Macbeth seems to show that he’s becoming paranoid, that the symptoms of his madness are growing, as he admits that he keeps a “servant fee’d”—he pays a spy—in each of the nobles’ houses (3.4.164). He no longer seems to trust his wife, whom he does not tell about his plans to have Banquo and Fleance killed; he no longer trusts the thanes who were once his fellows and friends; he no longer trusts Banquo, his one-time best friend. He is suspicious of all, and his suspicions seem to contribute to his growing madness.
In the end, Lady Macbeth takes her life—a result of her apparent mental illness—likely because of the burden of guilt she now bears. Her husband was once “full o’ th’ milk of human kindness,” and he’s become a monster, in part, as a result of her goading him into committing the first murder. Macbeth, for his part, has become a man who murders innocent women and children for no real reason and for no real benefit to himself. His ambition and pride have driven him to madness.