I have not liked a live-action Batman interpretation yet, save perhaps the 1966 film which exists apart from everything else due to the nostalgia factor The murdering-psychopath-Batman of the Nolan films is something best forgotten.
Does he murder in the Nolan films? I know he takes out the temple of the League of Shadows in Batman Begins, and leaves Raʼs al Ghul to die at the end. But he's far removed from a murdering psychopath. In the Dark Knight he doesn't even let the Joker drop to his death.
What Nolan does however in his trilogy, is question Batman's vigilante justice. Batman's acts, although motivated by good intentions, are unlawful and wrong. Batman seems necessary to take out the threats that Gotham faces, but the movies also ask the question: "at what cost?"
Save for Rachel 'I've been shoe-horned into this movie to provide a sub-plot for people who don't know who this 'batman' person is' Dawes.
I feel Rachel's role is a lot more important. She acts as a voice to teach Bruce (and his audience) that his views of justice are wrong. She helps explain how the criminal underworld of Gotham works. Her confrontation with Bruce after Joe Chill's death is one of my favourites in the movie. And I also really like the confrontation between Bruce and Falcony in that movie. Both Rachel and Falcony help Bruce realize that his view of the world is wrong, since he was born in a priveledged position. And it is through their words that both Bruce and the audience are set on this path of transformation. Note that after his transformation into Batman, Bruce does not kill Falcony. Nor does he kill Scarecrow.