Supposedly he kills because he's lost his way. Jason Todd or whichever Robin was killed by Joker, and he's gone apesh*t on crime ever since. Which is something that he does when a Robin dies. He goes ballistic on crime, and goes overboard, generally stopping at killing. But, BvS does a terrible job of pointing that out, and it's an arc that isn't resolved by the end of the movie, which makes Batman just look homicidal. He's killing killers so it's not as bad as it looks, but Batman should be better than that, and if he's going overboard on crime fighting, he needs to at least acknowledge that, or have Alfred tell him that, or have something to point out very blatantly to the audience that he's lost his way, and isn't just always a killer.
Edit: as for Snyder taking the heat, as the director, it's his job to look at the script and translate it to film. That includes seeing something like the "Martha" moment and realizing that it's a horribly tacky and thin excuse for Batman and Superman to find a balance, or at least make it work with proper dialogue that takes away tackiness. The director is the right person to blame for bad filmmaking 99% of the time.
I think I'll disagree here. Snyder messed a lot of things up, but Batman's arc wasn't one of them, imo. He had Alfred as the voice of reason and conscience throughout the entire movie, we are told from multiple sources that Batman is escalating his war and while Batman doesn't really have an overt moment of clarity and self-reflection like, say, Black Panther did in CA:CW, we can clearly see he's haunted and out of his element throughout the entire picture. I never felt the need for him to turn to the camera and explicitly state what goes on in his head like T'Challa does.
But I can see how that might have been insufficient. The problem is similar as with Clark himself in MoS, as I wrote about here before. Snyder puts too much stock in the audience translating their preexisting notions of these characters onto his adaptations, instead of doing the hard work of building them from the ground up. Clark suffered because of how choppy MoS was and also because of how the theme of his fathers grooming him into becoming Superman overshadowed his own personal character development and motivations. He was, part intentionally, part unintentionally, a blank slate for Jor-El, Pa Kent and the audience to project onto. And in BvS he's just a supporting character in The Batman Show 2015.
As I've said, I think that BvS's Bats is executed well overall, but true, the foundations of his character are shaky and fasttracked. We know that Bruce is at his lowest here, but we never see how that is different from the normal. We never saw this depiction of Batman. Snyder rushes straight into the territory that Nolan ventured in his final movie - after hours worth of careful character building. We know "Batman", we know how "Batman" should act, but we don't know this Batman. And that's the important bit. Snyder counts on the audience being generally acquainted with Batman's character and carrying that knowledge over to his movie and then just casually subverts everything we expect without properly elaborating on just what makes this version of Batman tick. That can be terribly jarring and misleading.
I really don't think Batman Begins 2.0 would be the solution, we don't need to see that all over again and like I said, overall I was satisfied with the result, but a movie with a premise like BvS was just a hard place to introduce a brand new version of Batman.