So, the Joker apparently didn't pull the trigger that put those holes in Jason Todd's Robin costume, after all.
According to the first act of Suicide Squad, someone else entirely confessed to the killing. As the characters are being introduced with stylish graphics introducing their names and rap-sheets, it turns out that Harley Quinn's profile says that she confessed to the death of Todd.
That seemingly kills the rumour that Todd remains alive (maybe), that he's not now the Joker (though Warner Bros might wish they'd done that so they can retcon a REAL Joker in later), and that Joker killed him in a Death In The Family sort of manner.
It also seems to fit given that Robin's suit is daubed with the words "Hahaha, Jokes on you, Batman", which was assumed to be Joker's calling card. In fact, the line was uttered by Harley Quinn on her first appearance in Batman: The Animated Series, and is actually - crucially - written on her bat in Suicide Squad.
Now, this doesn't necessarily mean that Harley did it, of course. It just explains why she's in jail. Given the fact that she proves with her plunge into the ACE chemicals that she will do ANYTHING for her Mr J, the idea of him getting her to confess to take some of the heat off him would absolutely fit.