One of Shakespeare's finest metaphors occurs in Act 3, Scene 4 of Hamlet when Gertrude expresses her opinion that her son is mad and he tells her:
Mother, for love of grace,
Lay not that flattering unction to your soul,
That not your trespass, but my madness speaks:
It will but skin and film the ulcerous place,
Whilst rank corruption, mining all within,
It is so characteristic of Shakespeare that he draws his imagery from common things with which we are all familiar. This metaphor describes something that happens to many people who put something on a cut and cover it up with a bandage, hoping it will heal but finding after a few days, when they dare to look at it, that it has only gotten worse. There is a transparent film over the wound, but underneath is an unwholesome-looking soft spot which Hamlet calls an "ulcerous place."
Earlier in that same confrontation Hamlet compares his uncle with his father as he shows his mother their two miniature portraits:
This was your husband. Look you now, what follows:
Here is your husband; like a mildew'd ear,
Blasting his wholesome brother.
In Act 3, Scene 3, when Hamlet decides not to kill Claudius while he is praying, he concludes his soliloquy with a striking metaphor:
This physic but prolongs thy sickly days.
Hamlet is suggesting that Claudius is a sick man because he is being eaten up inside by his guilty conscience, which is obvious in his unsuccessful attempt to pray. The "physic" Hamlet refers to is just his refraining from putting the king out of his misery with his sword at that time. There is a strong suggestion that Hamlet is enjoying seeing his uncle suffer and doesn't mind letting him suffer a while longer. Claudius has been unmasked by his terrified reaction to the play from which he fled. Hamlet understands him completely now. All his drinking is just an attempt to drown his guilt and his fears. He is a wretched man and only a counterfeit king.
There are many references to Hamlet's supposed madness as a disease. One memorable metaphor is spoken by Claudius at the beginning of the third scene of Act 4:
Diseases desperate grown
By desperate appliance are relieved,
Or not at all.
The "desperate appliance" Claudius has in mind is to have the English chop off Hamlet's head. When the king is all alone at the end of that scene he speaks his intentions aloud:
Do it, England,
For like the hectic [fever] in my blood he rages,
And thou must cure me.
There are numerous other references to disease and decay in the play described in previous answers by enotes contributors and accessible via the reference link below.