Heartbroken Badass: A non-romantic version. Jason's death and subsequent downfall into villainy left him broken and angry at himself for not being able to save Jason and leaving him with the wish that he could have done more for him if he had the chance.
Asskicking Equals Authority: He sets himself up as the drug kingpin in Gotham and rival to Black Mask by showing up at a meeting of the heads of the organizations under Mask's thumb and giving them a bag full of the heads of their lieutenants.
Cold Sniper: He kills three people from a significant distance without touching Batman or Nightwing standing to either side of them.
Combat Pragmatist: Though he has plenty of gadgets up his sleeves, he's not afraid to use anything in the environment to his advantage in a fight, and, as noted under The Unfettered, is willing to use any methods necessary to gain an advantage.
Complexity Addiction: His ultimate plan to get Batman to realize that sometimes Murder Is the Best Solution, specifically regarding irredeemable people like The Joker. Instead of just finishing The Joker off himself and using that as the ultimate symbol of his new pragmatic approach being more effective than Batman's idealistic approach, he goes to extra trouble to get Batman to abandon his Thou Shalt Not Kill rule and finish The Joker off.
Faux Affably Evil: He has a rather snarky, personable, and somewhat laid back personality, is cool under fire and logical with always a few tricks up his sleeves and sharp tongue to match. However, this belies his true nature of an angry, violent, young man with serious emotional and (possible) mental issues.
The Heavy: While he is the main antagonist and the one who drives the plot along for the majority of the film. It is The Joker who caused all that happened in the film to come about.
Irony: There's the fact that his actions are based around heroism, but are so brutal that he is a firm villain. But there is also a bit of symbolic irony in a bat family member becoming a new Red Hood, with the original being the original villain alias of who else but The Joker.
Knife Nut: After the guns, before the fists, he has a knife that can cut Batman's cables.
Knight Templar: Believes that everything he has done, every life he has taken, has been in the name of bringing order and stability to Gotham.
[To Batman] "Yes. You can't stop crime. That's what you never understood. I'm controlling it. You wanna rule them by fear. But what do you do with the ones who aren't afraid? I'm doing what you won't, I'm taking them out!"
The Reveal: Blood analysis shows that he's Jason Todd.
The Unfettered: Will use whatever methods necessary, no matter how brutal or violent, to achieve his end goal of "cleaning up Gotham."
Woobie, Destroyer of Worlds:He was once the partner of Batman and a former juvenile delinquent whose hard background, brutal death at the hands of the Joker, and subsequent resurrection drove him over the edge to become the murderous Knight Templar Red Hood.
Would Hurt a Child: His first scene shows the Joker beating a teenage boy nearly to death.
"Now, that was rude. The first boy blunder had some manners. I suppose I'm going to have to teach you a lesson so you can better follow in his footsteps. (beat) Nah, I'm just gonna keep beating you with this crowbar."
Ms. Li: Understood. We'll be taking further precautions at every transaction.
Mask: Screw that! Time he learned that this is a contact sport. We're going on offense. Rough up his business; something big! Something loud! When he shows up to shut us down, have a party waiting for him! And when I say "party" I actually mean a 'whole lot of people who are gonna KILL HIM!
Affably Evil: He states that he never resorts to violence or killing unless it's for a "greater cause" and is deeply remorseful that his hiring of the Joker was ultimately what lead to Jason's murder; his attempt to fix things via Lazarus Pit resurrection is what sets off the main plot.
The Atoner: After Jason's death, Ra's chooses not to oppose Batman again, and tries to resurrect Jason to try and repair the damage he inadvertantly caused.
My God, What Have I Done?: After his actions cause Jason's death, and again, after hearing of Jason's actions as the Red Hood, following Ra's raising him from the dead.
Unwitting Instigator of Doom: Twice. First, when his hiring of the Joker was ultimately what lead to Jason's murder. Second, when he did try to resurrect Jason to try and repair the damage he inadvertantly caused, but his actions only made things worse.