In many of Shakespeare's plays, the main character suffers from what classical Greeks called hamartia, loosely translated as a fatal flaw, and the most common fatal flaw is hubris, which basically means arrogance, the willingness to defy moral laws in order to achieve a goal that one believes he or she in entitled to.
For example, in Shakespeare's Macbeth, Macbeth's hubris, which leads him to believe that he should be king, causes him to murder his rightful king, Duncan. Unfortunately for Macbeth, because of his immoral act, his destiny is to be murdered himself in order for the rightful king to be restored to the throne.
The phrase "character is destiny," in the case of Macbeth, simply means that his flawed character, which is controlled by arrogance, leads inevitably to his destiny, which is death. Although "character is destiny" often has negative outcomes in Shakespeare's works, morally good characters sometimes are rewarded.