Joe Chill claimed to have been driven to mug the Waynes because of the desperation of the times (Gotham City underwent an economic depression because of an undisclosed action by the League of Shadows). Chill was arrested soon after he killed Bruce's parents. Years later, Chill underwent a hearing to be released from prison as part of a deal to testify against Gotham mob boss Carmine Falcone (with whom he shared a prison cell with) in exchange for parole.
During the hearing, Chill claimed to regret his crime. After the hearing despite the GCPD's presence, Chill was killed by one of Falcone's assassins, who posed as a reporter as he left the courtroom. It was later discovered that Falcone had bribed Judge Faden of Chill's case to make the hearing public and to bring Chill out into the open. Bruce, who waited outside the courtroom with a gun of his own, was thus deprived of his own chance of revenge. Bruce's lost chance of killing Chill himself helped him realize what justice really was, as well as his memories of a gun taking his parents' lives that brought him to his rule that he would not kill. Bruce later confronted Falcone, who taunted him by saying that Chill had bragged that Thomas had "begged like a dog" before his death (that version of events was a lie as we saw the murder of Bruce's parents earlier in the film. Also, Bruce witnessed his parent's deaths).
Overall, Joe Chill was depicted as a tragic character in the film, since during his mugging of Bruce's parents, he's shown to constantly be shaking from what he was doing. Also, later, when Ra's al Ghul informed Bruce that the League of Shadows had caused Gotham's economic depression, he commented that Bruce's parents were "gunned down by one of the very people they were trying to help."
Bruce's final confrontation with Chill slightly echoed his confrontation with Chill in the comic storyline, Year Two: Fear the Reaper, where Batman was forced to work with Joe Chill to stop a new vigilante who killed criminals. Their 'alliance' culminated in Batman taking Chill to Crime Alley after the Reaper's apparent death, where he unmasked in front of Chill, the two men fighting for a time before Chill was forced to his knees while Bruce pointed Chill's own gun at his head. However, the Reaper appears and kills Chill while Bruce is still debating whether he can actually pull the trigger. In both Begins and Year Two, it is thus left ambiguous whether or not Bruce would have actually killed Chill, as someone else beat him to it before he could make a decision either way.