Let me try to get right down to what I see as the character that I think people so often get wrong:
He gets to kill. ONCE. After that, he basically has to quit. (I am of the opinion that he KILLS THE JOKER in Dark Knight and has a bit of a psychotic break because of it, thus hallucinating the Joker "breaking his own back." Look at the word balloons and explain to me how a person could twist their own back enough to snap it. JOKER DOES NOT KILL HIMSELF. Batman kills the Joker and can't admit it to himself.)
Killing is the sin that created him; he won't repeat it (except the once, and that's probably the Joker, TBH. And doing so means the Joker has won in corrupting him). He turns criminals over to the system hoping that the system will work. It's a bit naive, but it's also showing his overall faith in humanity.
He fights crime because he has a COMPULSION to seek justice. This is shown VERY WELL in the Barr run (Detective 569-582 or so--the first seven issues feature WONDERFUL Alan Davis art). He's NOT out for vengeance. It's not so much that he wants to hurt criminals as much as it is that he CANNOT ABIDE THEIR PRESENCE. This is the one problem I have with the DCAU version, because he claims, "I am vengeance!" in a Season One episode. Vengeance would be getting Joe Chill and killing him. That's NOT the point. He doesn't want this to happen to ANYONE ever again.
He builds a family around him. He wants it very badly, but he doesn't really know how to manage it because he never had a family growing up. The whole "Jason was a good soldier" thing from Dark Knight REALLY grates on my nerves. I allow it in THAT universe, but I HATE it in most other universes. It's a really extreme position on Batman.
Bruce Wayne tries to rehabilitate the city. He offers jobs to ex-cons, he does education programs, etc. He's the "day" part of Batman and probably does more good in the real world than a crazy vigilante could ever do.
He is a DETECTIVE first, and a fighter/ninja/superhero second. SOLVING CRIMES is most important to him. He won't EVER allow one to go unpunished, but he does his best work with his mind, NOT his fists.
He's fundamentally a GOOD PERSON, but he's somewhat broken because of his trauma. I've always liked the idea that children are NEVER scared of Batman because he's only scary to guilty people. The Batman of Batman V Superman is totally, 100% wrong as a "mainstream" Batman, IMO.
He's HUMAN. He can be beaten in a fight. He makes mistakes. He has to be careful or he'll be dead. The whole "Batman can't be beaten if he has time to prepare" idea is STUPID to me. Profoundly so. Batman vs Superman ends with the Batstain IF Superman isn't LETTING HIM WIN. He's incredibly formidable, but part of the reason he needs the JLA is because he's NOT PERFECT. Wonder Woman is a better fighter. Superman is a better inspiration. The Flash is a better scientist. Cyborg is better with computers. Hal Jordan has a stronger will. Aquaman has WAY more sushi. But in most instances, Batman is #2. Especially with the sushi; Catwoman can't resist it. ME-OW!
KGBeast, Lady Shiva, the Sensei, and others are better fighters. Ra's al Ghul is a better leader and strategist. Any Olympic athlete is better at their sport than Batman. But he's the BEST DETECTIVE THE WORLD HAS EVER SEEN. Him vs. Sherlock Holmes results in a 3-2 Batman win in any mental contest.
In a just universe, he dies happy. Dick can take over for him, their relationship is solid, and Gotham is a better place because of his life's work. BUT HE DIES, because he's human. And that's okay.