Throughout the play, Hamlet claims to be feigning madness. Do you think this true or is Hamlet actually insane?

Hamlet is upset, but not insane.  

Hamlet’s behavior is erratic, but calculated. Although his speeches do seem strange at times, and the way he behaves can sometimes seem bizarre, he is not actually insane.  One way you can tell this is that he goes through a great deal of trouble to make everyone think he is crazy until Rosencrantz and Guildenstern show up.  He lets them, and us, in on the secret. 

HAMLET

…You...

Hamlet is upset, but not insane.  

Hamlet’s behavior is erratic, but calculated. Although his speeches do seem strange at times, and the way he behaves can sometimes seem bizarre, he is not actually insane.  One way you can tell this is that he goes through a great deal of trouble to make everyone think he is crazy until Rosencrantz and Guildenstern show up.  He lets them, and us, in on the secret. 

HAMLET

…You are welcome: but my
uncle-father and aunt-mother are deceived.

GUILDENSTERN

In what, my dear lord?

HAMLET

I am but mad north-north-west: when the wind is
southerly I know a hawk from a handsaw. (Act 2, Scene 2) 

Hamlet wants his childhood friends to know that he is not actually crazy, but just pretending.  His crazy act is part of the plan to get revenge on Claudius for his father's murder, but he feels that they should know the truth.  He either feels embarrassed because they are seeing him like this or trusts them enough to let them in on the secret. 

Hamlet's intelligent and heartfelt pondering of the meaning of life and death, his famous “to be or not to be” speech, is not the babble of a crazy person.  It is the introspection of a young man full of grief and self-doubt.  It is a person who is hurting, but still able to look into the abyss and ponder the meaning of our existence.

Another way you can tell that Hamlet is not crazy is his reaction to Polonius’s death.  While he definitely intends to make his mother continue to doubt him, there is some sincerity in his words.

Thou wretched, rash, intruding fool, farewell!
I took thee for thy better: take thy fortune;
Thou find'st to be too busy is some danger. (Act 3, Scene 4)

Hamlet feels bad about killing Polonius, despite the terrible mind games he plays with everyone else over his body.  He uses the untimely death of the man as a pawn, but he did not mean for it to happen and he grieves for Polonius.