Wasn't the Dark Knight against killing criminals?

This question has arisen quite a lot since Dawn of Justice came out, and while I think it's a fair criticism of Zach Snyder's Batman, the question is usually framed in a way that has a few problems: the idea that this Batman kills for no reason, the idea that the movie never explains why this Batman "changed", and the idea that Batman in general doesn't kill people.

First of all, it's important to keep in mind, as with every other reboot of Batman, that this is a brand new character, created by DC Extended Universe "creative director" Zach Snyder and his various writers for Dawn of Justice and it's sequels. Obviously, he is using the name and likeness and general backstory from the DC comics character his movies are based on. But just like Kal El was given a new personality, Bruce Wayne has his own new, original personality in this movie. So it's not valid to say that his behavior is inconsistent with any prior actions or philosophy, because there are none. The DCEU Bruce Wayne has never been in any other movie before this, so his actions in Dawn of Justice are all we have to go on.

Second, the movie tries (but maybe doesn't succeed) to get across the idea that Batman was not always like this. Alfred repeatedly chastises Bruce for what he's doing and how he's doing it, and there are multiple conversations about the idea that Gotham ultimately corrupts all good men. Though they don't go much into the backstory of Batman yet, the very strong implication is that The Joker has done something incredibly horrible -- probably involving the death of Robin, who's suit we see in the movie -- that even led Bruce to retire from crime fighting. It was only the arrival of Superman and the destruction he caused that led Bruce to go back into business, and his experiences have made him dark and jaded and angry. So, as Batman, he's now doing the wrong thing because he's allowing his anger to overcome him.

The final scene in the movie, between Bruce and Diana, practically comes out and says this. Bruce claims that he had let Clark down with his actions, but he's going to do better, and make it up to him. Snyder seems to be leading Bruce on a process of character development where he recognizes the danger and fallout of his previous behavior, and strives to do better.

Third, and possibly most important, Batman is not always averse to killing people. Batman in Dawn of Justice may be more overt about it, and they may show it more obviously on-screen, but in pretty much every Batman movie to date Batman has done things that can't possibly have left the target alive, up to and including firing missiles at people from the Batmobile. Batman, in the source comic material, has a "moral code" against killing people, but a lot of that is momentum stemming back to the Golden Age of comics, and particularly during the dominance of the CCA, when the hero would never be allowed to hurt someone on-panel. Even then, there has been more than one incident in the comics -- as far back as pre-CCA-era 1940's -- where Batman clearly kills someone in cold blood.

KutuluMike answered it quite well but here is what Zack Snyder said himself :

I tried to do it in a technical way. There’s a great YouTube video that shows all the kills in the Christopher Nolan movies even though we would perceive them as movies where he doesn’t kill anyone. I think there’s 42 potential kills that Batman does! Also, it goes back and includes even the Tim Burton Batman movies where this reputation as a guy that doesn’t kill comes from.

So, I tried to do it by proxy. Shoot the car they’re in, the car blows up or the grenade would go off in the guy’s hand, or when he shoots the tank and the guy pretty much lights the tank [himself]. I perceive it as him not killing directly, but if the bad guy’s are associated with a thing that happens to blow up, he would say that that’s not really my problem.

A little more like manslaughter than murder, although I would say that in the Frank Miller comic book that I reference, he kills all the time. There’s a scene from the graphic novel where he busts through a wall, takes the guy’s machine gun…I took that little vignette from a scene in The Dark Knight Returns, and at the end of that, he shoots the guy right between the eyes with the machine gun. One shot. Of course, I went to the gas tank, and all of the guys I work with were like, ‘You’ve gotta shoot him in the head’ because they’re all comic book dorks, and I was like, ‘I’m not gonna be the guy that does that! - { Source: heyuguys.com}

Here is that video:

Even KutuluMike's answer mentioned this video.

And below is the full interview of Zack Snyder: