Really? The Joker.... I dunno. I mean, he's crazy, but still...
Plus the Joker's Red Hood days were actually prior to his chemical bath. He's too attention-loving, spotlight whore to cover up himself with a hood nowadays.
Plus the whole "punishment fits the crime" is more like the vigilante who kills Jason style. No way the Joker would kill someone for killing. He would just thoroughly encourage more killing.