What makes Hamlet “the first true tragedy of character?”
The first step to answering your question would be defining your quote. You state: "the first true tragedy of character," and this can be read in several ways.
1. By "tragedy of character", you can argue that Hamlet's character in and of itself is tragic because it is fatalistic. Through this analysis, we comprehend that Hamlet delineates from the issues at hand - his father's murder, his mother's remarriage - to consider whether he wants to live or die. "To be, or not to be, that is the question" resonates with all readers because we comprehend the internal struggle that Hamlet faces between "suffer[ing]" through life, or "sleeping," committing suicide. This is a tragedy of character because Hamlet is in a state of constant depression that we know will never end well.
2. By "tragedy of character," you may also argue that Hamlet, as a character, never completely accomplishes the quests he creates for himself. His first plan is to avenge his father's "most foul murder" by killing the assassin, his uncle and the new king, Claudius. While the play ends with Hamlet murdering Claudius, Hamlet himself dies within minutes thereafter because he was stabbed by Laertes, thus never really 'enjoying' his revenge. He also questions whether he should live or die, constantly debating whether suicide is the way out, and he never comes to a conclusion or determination about which is better. He wants to repent for how he treated Ophelia; however, after he decides to do so, he is too late because she is already dead.
Thus, Hamlet may be considered "the first true tragedy of character" because everything about his situation was deemed hopeless from the beginning of the play.