Hamlet is depressed from the start of the play through act 3, scene 3 because of the death of his father and the hasty remarriage of his mother. His depression is aggravated by the anxiety he experiences when he meets his father's ghost and learns that he was murdered by Claudius. The ghost also demands that Hamlet avenge his death.
Hamlet's possible tragic flaw is his indecision, though one could also argue that he is behaving ethically and prudently in testing whether the ghost is telling the truth about Claudius before he kills him. Hamlet is in psychological agony as he feels caught between the filial obligation in his society to avenge his father's death and his need for clarity on Claudius's guilt or innocence. Hamlet has frequent suicidal thoughts, though the possibility of an afterlife keeps him from killing himself.
By the end of act 3, scene 2, Hamlet has proof of Claudius's guilt in his reaction to The Mousetrap play. Nevertheless, when Claudius is at his prayers and Hamlet could easily kill him, he hesitates and decides not to, rationalizing that it would not really be revenge if Claudius had just confessed his sins and would go to heaven, while his own father suffers in purgatory. At this point, however, we might wonder if Hamlet is simply being indecisive and coming up with excuses not to act. This reading is borne out as we as an audience overhear Claudius showing no repentance for all that he has done but instead expressing that he is glad he is king and Gertrude's husband.
We've also witnessed Hamlet treating most of the people he is in relationship with poorly. He is cruel to Ophelia and treats her in a way that confuses her, one moment seeming interested and the next rejecting her. He is cruel to Polonius and the courtiers. The only person he treats well is Horatio.
At this point in the play, it would be permissible to think that Hamlet has not experienced much growth or development as a character, is stuck in self-absorption and indecision, and has some maturing to do.