Even as the boy wonder partner of Batman, Jason Todd was a brash and morally complex individual. Unlike his mentor, Jason was far more relaxed about the idea of dispensing justice however means necessary, and is even teased to have pushed a serial rapist off a balcony in 'Batman' #422.
His death at the hands of the Joker remains one of the most brutal and infamous moments in all of comic book history, and without a doubt pushed Jason much further towards being a violent anti-hero once he was resurrected.
Though he still remains fixated on bringing the wrongdoers in society to justice and has proved to be a valuable ally to Bruce Wayne on more than one instance, there's no denying that Red Hood's means of dispensing said justice is a little on the grey side. He uses a whole range of weaponry rarely ever seen with Batman and, crucially, breaks Bruce Wayne's one rule of not killing anyone quite happily.
The years of emotional trauma and living in the shadow of the Bat for so long might be an understandable reason, however Red Hood's instabilities and lack of moral code do really make for one violent good guy.