Lewis "Lew" Moxon is most famous for hiring Joe Chill to murder young Bruce Wayne's parents, thus making him directly responsible for Batman's existence.
Golden / Silver Age version
At least ten years prior to their murder, Dr. Thomas Wayne and his wife attend a costume party - to which Dr. Wayne wears a bat-like costume. Wayne is taken from the party, by gun-point, to meet racketeer and bank robber Lew Moxon. Moxon orders Dr. Wayne to remove a bullet from his shoulder. Wayne carries out the operation before overpowering Moxon and his men.
Moxon is arrested and sentenced to 10 years for armed robbery. As he is taken away he swears revenge on Thomas Wayne. Ten years later, a free Moxon informs Thomas Wayne that he will get someone else to extract that revenge (that someone being Joe Chill). It is suggested that Chill was told to leave young Bruce alive so as to provide an alibi for Moxon.
Years later Batman learns of Moxon's involvement in the murder of his parents, by which time Moxon is operating a Billboard blimp business. Batman's first attempt to convict Moxon for the murder of his parents fails, as Moxon is now suffering from amnesia and is able to pass a lie detector test. Batman determines that Moxon's blimp business must be a cover for illegal activities and continues to trail the gangster. Eventually, after uncovering Moxon's illegal activities, Batman confronts the criminal while wearing his father's costume (his own had been ripped during an earlier fight). On seeing Dr. Wayne's costume Moxon suddenly remembers what he had done. Thinking Batman was actually Thomas Wayne's ghost, Moxon panics and runs out into the street, where he is hit by a truck and killed.
Modern Age (Post Zero Hour) version
Lew Moxon invites Bruce Wayne to a party. Wayne recognises the name and investigates. Moxon, it transpires, is an aging mob boss with failed political aspirations, returning to Gotham City after several years in exile. Wayne attends the party, at which he is introduced to Moxon's daughter, Mallory Moxon. In actual fact Bruce and Mallory had been childhood sweethearts. The Moxons and the Waynes had met at a holiday resort when Bruce was seven or eight. At the party Wayne realises that Lew Moxon is the target of an assassination by Deadshot and also that Moxon has employed Philo Zeiss as his bodyguard. Earlier, Zeiss had killed Bruce Wayne's friend Jeremy Samuels. Later, Zeiss informs Batman that he carried out this murder in order to hurt Wayne, because of something that Thomas Wayne had done to Moxon.
Moxon survives Deadshot's assassination attempts (though he now requires the use of a wheelchair), and Batman discovers Mallory is now actively in the family business. Ultimately Batman discovers that Thomas, Martha and (a very young) Bruce Wayne had attended a costume party (to which Dr. Wayne wore a Zorro costume) which was also attended by Moxon. At the party, Angelo Berretti, an 'employee' of Moxon's, told Dr. Wayne that a man's life was in danger. Dr. Wayne made Berretti promise his safe return before departing. Dr. Wayne was informed that Moxon's nephew required a bullet to be removed from his shoulder (following a failed armed robbery). Dr. Wayne performed the operation. Dr. Wayne refused to take any money, and started a fight with Moxon. Following this, Moxon had ordered the murder of Thomas Wayne. However, this was not carried out because of Berretti's promise.
Years later, Batman questions Angelo Berretti as to whether Moxon had been involved in the killing of the Waynes (which occurred a few months after the costume party). According to Berretti, Moxon was not involved. After Batman accepts Berretti's word and departs, Berretti holds his hand to his face and utters an ambiguous "Oh, thank God". Moxon's involvement in the killing of the Wayne is, therefore, left to the readers imagination.
Moxon is ultimately killed by Zeiss at a meeting of Gotham's gang bosses (arranged by Stephanie Brown). He had employed the costumed villain Hellhound as a bodyguard, but Zeiss easily defeated him and proceeded to murder Moxon, killing Hellhound in the process. This event occurs at the commencement of the Batman: War Games story line.
Place in Continuity
During the Golden Age of Comic Books, Lew Moxon was established as the man who hired Joe Chill to kill Bruce Waynes parent's.
During the Silver Age of Comic Books and Bronze Age of Comic Books it can be assumed that the Lew Moxon story remained canonical for tales set on Earth Two. For several years no mention is made of Moxon's role regards the origin of Batman on Earth One - that is until the Len Wein-written mini-series, The Untold Legend of the Batman (published in 1980) re-introduced the Lew Moxon story, exactly as it was told in 1956.
During the post-Crisis era no mention is made of Lew Moxon - therefore Moxon's canonical status during this time is uncertain. In Batman: Year Two, Batman confronts Joe Chill, much as he did in the Golden and Silver Age continuities. However, just as in previous continuities, Chill makes no reference to Lew Moxon before his demise.
During the post-Zero Hour era, while Lew Moxon does appear in several issues, Joe Chill does not.
After the events of Infinite Crisis, Joe Chill is again known to be responsible for the murder of the Waynes, and he was arrested for the crime on that same night. No mention has so far been made of Lew Moxon, suggesting again that he was not involved. This is further backed up in Grant Morrison's Batman #673, in which Batman confronts Chill and discovers the crook acted alone.
In other media
Lew Moxon appeared in an episode of Batman: The Brave and the Bold called "Chill of the Night!" Here he is dying of presumably natural causes and is visited by Batman, disguised as a clergyman, while in the hosptial. Moxon admits to some of the things he has felt bad about over the course of his life. He also confesses that he had ordered the murder of Thomas and Martha Wayne, due to the fact that he had been thrown into Blackgate Penitentiary because of a past event involving the young doctor. Moxon regrets the murder of Martha Wayne, something that was not part of the original plan, saying he thought it was wrong to leave young Bruce without a mother. Batman demands the name of Moxon's hired killer several times, but Lew avoids the question. Finally, with his dying breath, he gasps out a single word: "Chill". This results in Batman reopening the Wayne murder case.