Joe Chill is the name assigned to the murderer of Bruce Wayne's parents. Chill is best treated as a non-character. An anonymous prime mover in the story of Batman. "Joe Chill" may or may not be his real name. He's basically a nondescript mugger who represents all the criminal scum who prey on the innocent.
In most accounts, Chill is represented as a simple mugger. Perhaps an alcoholic or addict who'd so forsaken his humanity that he'd do anything to get his next fix. It's not clear whether or not Thomas Wayne actually tried to fight Chill off, or was merely trying to move young Bruce out of the way, but something spooked Chill and prompted him to kill both Thomas and his wife right in front of their little boy. Chill then dissappeared into the darkness.
There have been a couple of attempts to flesh out the Joe Chill character and the story of the Wayne murders. During the awkward phase between the golden and silver ages, one storyline told us that Chill was actually making an intentional hit on Thomas Wayne due to Wayne's own crimefighting actions, and that Martha Wayne had died of a grief-induced heart-attack rather than gunshots. The 1989 Batman movie had the Waynes murdered by a criminal who would later become the Joker.
Ultimately, such elaborations are unwarranted and best ignored. For the sake of his role in Batman's origin, Joe Chill cannot be an individual. He can not be a person who Bruce Wayne can track down and confront, as this would allow closure on the Wayne murders, and negate Bruce's need to fight a personal war as Batman. Because Bruce will never identify and confront Joe Chill as an individual, Chill becomes the embodyment of all criminals. So, to Bruce, every mugger is Joe Chill. Every Criminal is subject to his vengence.
In truth, Joe Chill probably died in a gutter years before Bruce Wayne first donned the cape and cowel, never having a clue as to what his actions had unleashed upon the criminal element.