In what ways did Hamlet manage his identity? For what purposes? How did he present both private as well as public identities?

Hamlet manages his public identity by pretending to be mad.  He tells Horatio that he is going to act insane in public so as to deflect any possible suspicion that Claudius might feel as Hamlet goes about his investigations into his father's death.  He then proceeds to act like a mad person in front of Claudius, his uncle and step-father and the man he believes murdered his father, as well as Polonius, Claudius's closest advisor, because he doesn't trust either one. 

He also begins to act crazily in front of Ophelia, his former lover, because she is the daughter of Polonius; she has taken her father's instruction and distanced herself from Hamlet, the man who genuinely loves her.  Hamlet seems to feel somewhat betrayed by her treatment of him, but it is also possible that he wants to protect her from whatever fallout might occur as a result of his investigations into Claudius, and so he keeps the distance between them as a way of keeping her safe from the king. 

Privately, however, with Horatio, Hamlet's best friend and the only person he really trusts anymore, he is the same as he ever was: clever, witty, thoughtful, and so on.  This is one of the ways the reader knows that the persona he presents to the rest of the characters is, in fact, an act.