I can't see the anti-hero idea being a thing at all.
Like batman, his driving force is the feeling of weakness: BM when his parents are killed but he's only a kid, so he makes himself strong and cleans up criminals. Joker is weak as an adult, though, and his beef is with ordinary people because they show him little respect or sympathy. We might feel sorry for the treatment he faces, might wish for him to get some form of revenge, but we won't empathise with how far, and where, he takes it. Joker can't make himself strong physically, doesn't have wealth, doesn't have interpersonal skills, so finds his strength in causing fear and repulsion: he weaponises the only emotions that people have previously shown toward him when he was just an ordinary weirdo.
Just what I'm getting from the trailer. I don't read comics.