In "Hamlet," what is Hamlet's state of mind at the end of Act III? Is he only pretending to be mad?

If Hamlet is not mad, he is, by his own declaration "thought-sick."  Clearly, with all that has happened, Hamlet suffers from severe melancholia. And, research has shown that psychiatrists today often diagnose patients who are realistic with clinical depression.  Hamlet is such a person. For, he is sickened by the reality that his mother has committed an act of perverseness by marrying her brother-in-law who has killed his father: "reason panders will" (III,iv,88).  Repulsed by her lack of shame, he tells his mother,"And would it were not so, you are my mother (III,iv,15).

Justly distrustful of Claudius, in Act IV Hamlet feigns madness  with the king by answering him in riddles.  By doing so, he exemplifies the first line of a poem by Emily Dickinson:  "Much madness is divinest sense" since Hamlet wishes to keep the king off guard so that he can later watch his reaction to the trap he sets with a play.