Batman has existed for over 75 years and the character has had many iterations. We've had numerous comic book versions and many different movie versions. Most recently, Ben Affleck has taken up portraying the superhero on the big screen. His version of the Dark Knight was met with praise and complaints. Some say he was the only positive aspect from Dawn of Justice, others say he was the worst when he was so torn up by hearing Superman utter the name, "Martha." However, this version of Batman draws upon source material while also shaping a new version of the character. To start, let's look at a history of Gotham's Dark Protector.
ORIGINS OF THE "KILLING" BATMAN
Batman first appeared in Detective Comics #27 after being created by Bob Kane and Bill Finger (Bill Finger actually had the ideas for most of Batman's designs and even wrote the initial script for the first appearance). Anyway, this version of Batman used guns and killed a few people. He justified it in the stories, but it was the vigilante form of justice Batman was known to carry out.
This already is similar to the Batman in Dawn of Justice. Killing or using extreme measures of justice that is technically illegal, similar to how the plot played out in The Dark Knight. If Batman killed or used methods that were too harsh, police would react as they did in The Dark Knight and Dawn of Justice. Police chased Batman relentlessly as soon as he came back in The Dark Knight Rises. In Dawn of Justice, a police officer tries to shoot Batman, although he could have just been startled since he also almost shot his partner. But later, the media is portraying Batman as a vigilante, although a more extreme version since he is resorting to branding criminals. So just in the first appearance of Batman, we see some similarities.
The early Golden Age version of Batman resorted to brutal methods that often did result in criminals dying by his hand. He kicks a man into a vat of acid, shoots others, hangs some by his plane (as pictured above) and breaking some criminals' necks. However, this version only lasted a few years. The end was marked in Batman #4.
Bob Kane and Bill Finger were still developing the character and they decided that their character will not kill. The early Golden Age version of Batman almost a completely different version. He didn't even have his origin of having his parents gunned down in front of him and deciding to become Batman until after a few issues of the comics had already been created. The core of the Batman character did not exists until Detective Comics #33, when we see the murder, the vow, and the bat crashing through the window. Until then, the identity of Batman was never set in stone.
Still, Batman has killed on numerous occasions since he has had his character developed. He's dodged gunfire to have it hit criminals, which isn't exactly him killing; he locked KGBeast (who's real name is Anatoli Knyazev, the terrorist working with Luthor in Dawn of Justice) inside a sewer to let him die; and Batman has straight up killed Ra's Al Ghul by flying him into a giant laser and letting the ashes be sucked into the vacuum of space. So to end this, Batman has killed in many versions of his character, including the more recent Arkham video games.
WHY BATMAN KILLS IN DAWN OF JUSTICE AND MOTIVATIONS
As for the reason why Batman is the way he is, times have been tough in Dawn of Justice. He has been in action for over 20 years by the time we see him and he feels as though he has accomplished nothing. Criminals still remain in Gotham, he has quite the Rogue's Gallery, as we can see in Suicide Squad, and to top it off, Robin was killed by the Joker. Batman is in pain in Dawn of Justice.
I'll tread on some thin ice here: Any person would want to kill after losing a loved one. We're angry, sad, depressed, and many other feelings. So when Batman made it his mission to become a vigilante and try to suppress the crime in Gotham, he didn't want anyone to lose a loved one again. He doesn't want an innocent life to be taken. So he decides criminals can die if they get in the way of his mission. Fast-forward to the moment Batman hears Superman say, "Martha," and he loses it a bit and wants to know how Superman knows that name. Upon finding out that Martha is Superman's mother, they become besties and Batman goes to save Martha.
Reviews from people across the internet screamed about that and memes were created. However, Batman is crazy. He's a man at his peak, Olympic-level athlete dressed as a Bat that goes around beating up criminals and even has the guts to fight an alien that destroyed half of Metropolis in a fight. And people are complaining about the Martha incident. In that scene, it's not the name being the same that makes Batman and Superman buddies. It's the realization of who Batman is as a character. Upon hearing the name Martha and realizing it is Superman's mother, Batman realizes that Superman has parents. He is about to kill an innocent that most likely will result in the death of another innocent, Martha. It's going against everything Batman swore he would defend.
Batman in Dawn of Justice has no issue with killing those that get in his way, but when it comes to his mission, he will save the innocent. I once wrote a report on Batman: The Ultimate Evil by Andrew Vachss, and Batman's whole mission was to save children who were in sex trafficking. When it came to it, he let one of the men behind it all die. It's simply part of the Batman that was not geared directly for kids. The animated series never had him kill, but almost every other version has at some point. Batman has killed KGBeast, Darkseid, Ra's Al Ghul, Joker (up for debate based on The Killing Joke), and many other criminals.
Ben Affleck is quoted as saying:
He’s on the verge of being swallowed up by the anger and the rage that we see haunt this character in the other manifestations of it. But this guy is further down the line and has become more embittered and cynical.
While embittered and cynical is not Batman was we see him, it could be this version. The one most people see is the selfless hero who patrols Gotham City. Batman is a man putting his life on the line to battle the criminal underworld. However, the Dawn of Justice version is embittered from having put so much into his mission, but sees very little results. And of course he would become cynical if he saw no results because he trusts no one and believes that he is the only one who even has a chance to do anything, but even then he has doubts. The Batman of Dawn of Justice is tired, but feels his mission is not over. After 20 years of hardly any results, he feels the way he does and would resort to more extreme measures, although still sticking to his mission to protect/cleanse Gotham.
THE WORLD'S GREATEST DETECTIVE?
This bothered me slightly in Dawn of Justice. Batman is the world's greatest detective or at least is capable of doing very well thought investigations and planning. He is a master tactician and can prepare for many scenarios. Yet, he struggled in Dawn of Justice. Granted, this version is different, but he failed to investigate quite a few things: the explosion at the Superman hearing, the checks from Wallace Keefe, the White Portuguese, and not knowing Superman's identity before Lex Luthor. The movie implies that blind hatred drove Batman to not investigate, which I am okay with accepting. However, given that this is an emotional driven Batman that has been through quite a bit, I understand the direction the story was taken. But given that Batman has been operating for over 20 years and is still alive, he should be a better detective.
WHAT'S NEXT FOR BATFLECK?
To be honest, I felt that Batman actually developed in Dawn of Justice. Everything stated is debatable, although I feel as though it should be debated. This is a new version of a memorable character that had a very controversial casting. I grew up with Batman watching the cartoons and even the movies before I was old enough to actually be watching them (You know, Batman killing the Joker by attaching a gargoyle to his leg). This version of Batman was new and different and I really enjoyed the new direction. The ending scene where Batman confronts Luthor in his cell made me believe the character was developing, as Batman chose to not brand Luthor. It was a change, although a small one. And in the DCCU, this small change will echo across all of the planned movies.