No one knows who the Joker really is. Little can be confirmed regarding his early life before he turned to a life of crime. Despite his capture, no traces could be found on his fingerprints, dental records, or DNA matches against the GCPD's databases.
The Joker's own testimony, while normally quite true when it came to carrying out threats, seemed at times contradictory, and he was known to lie at times when describing past events in his life. One of his anecdotes told of an extremely abusive and alcoholic father, stating that after attacking his mother with a knife, the blade was next turned on the young man, creating his mutilated smile. If the statement was at least partially true, it could account for both mental and physical scarring that would give rise to the sociopathic and homicidal clown.
The catch is that the story conflicts with another one the Joker told later, stating that he once had a wife. Locked in a debt with loan sharks, enforcers came and cut his wife's face. In a desperate effort to assure his wife that he did not care about the damage to her appearance, the future Joker took a razor to his cheeks to produce his Glasgow smile. But the disturbing image instead caused his wife to leave him, possibly damaging his psyche.
It is widely speculated that one story or the other is a lie, or that perhaps both are falsehoods. It's also not impossible that both stories could actually be somewhat true, in that the Joker hinted to scarring on his left side when his knife was pressed to Gambol's mouth, while the scene with Rachel Dawes alluded to damage on the right side. The Joker's body language during these stories would suggest that certain truthful elements were in both tales, but that much of it was a fabrication. The powerful contradictions however, leave the Joker's background as is: completely unsolved. To enhance the popularity and mystery of the Joker's character and personality, his creators may never wish to reveal, or create, a true backstory explaining many aspects of his origins as his criminal reinvention as the Joker, including his cheek scars.
Possibly the most telling truth about his past is a comment he made to the Manager of Gotham National Bank. When asked what he believed in, the Joker stated that he believed "whatever doesn't kill [a person] simply makes [them] stranger". This would seem to suggest that a violent attack or other incidents he suffered through helped to damage his mental health, and if attacked, possibly inflicted the cuts that would become his cheek scars.
However, what is more probable is that the Joker origin stories, while perhaps containing some truth, more importantly display one of the few consistencies in his character. The Joker asks on three occasions to different individuals (Gambol, Rachel Dawes, and Batman), "Want to know how I got these scars? ...," as a prelude to action. In Gambol's case, the Joker tells his story and then kills him. With Rachel Dawes, the Joker ascertains that she might know Harvey Dent's location threatens her, but is interrupted by Batman before he can do anything to her. In the third case with the bombs on the ferries, the Joker is interrupted by Batman before he can blow up the ferries and even before he can tell another story about his scars. Perhaps this is homage to Jack Nicholson's Joker, who says before shooting Bruce Wayne, "Have you ever danced with the devil in the pale moonlight?" a line recited by the Joker/Jack Napier before killing or exacting harm on his victims including, in Bruce's recollection, Wayne's parents.
It is also possible that the Joker's scar stories, while containing some truth to his past, are also tailored to the person whom he's telling/threatening. In Gambol's quest, the question "Why so serious?" may be directed at Gambol himself, based on his excessive rage at the Joker during his meeting with the mob bosses. With Rachel Dawes, the story about his wife, whom the Joker says to Rachel was "beautiful, like you," and her resultant fear of him when he deliberately mutilated himself, might be directed at Rachel herself, who is visibly frightened by the Joker. The stories could thus be seen as frightening jokes, funny to the Joker, about his past related to current situation.
It is also possible that the Joker himself is unaware of his true origins. Some psychological profiles of The Joker, indicate that he is so insane that he literally reinvents both his own psyche and history on a daily basis. It is therefore possible that, while neither story is true, he genuinely believed them both as he told them.
Given that he might not know the true origins, the versions could have been partially true. For example he could have at least had an abusive father. This was supported when he attacked Bruce Wayne's party, searching for Harvey Dent, when an elderly man stated that he wasn't intimidated by thugs. The Joker then grabbed him and said that he reminded him of his father and that he hated him. Also since the father story may have been somewhat true, we can assume that he also had a wife who left him, most likely for similar yet different circumstances. In conclusion, as stated before, he may just twist his psyche to make a past which he feels fits best. So he took events of his life and used them as a reason for his scarring.
Also assuming this theory is true, his true scarrings could have come from hints from previous comments. When he was in MCU, he mentioned ripping off mob dealers and as stated before he told the bank manager that "whatever doesn't kill you, simply makes you stranger". So the Joker could have messed with dangerous people and they retaliated by scarring his face.
So we can only assume that his life went this way. He was raised by an abusive father whom he grew to resent. Later he married an attractive woman, who for unknown circumstances left him. And finally he ripped off mob dealers who retaliated by scarring him (this also could have been the reason why his wife left him). Of course though these are only theories and everything the Joker said could have in fact been merely lies or at best delusions. On the other hand they could be peices to the puzzle of the Joker's mysterious origins.
Considering that the Joker's image is that of contradiction and chaos, the stories that were told about his past always involved him being the victim, also imply he killed his father and his wife for hurting him, or made them up while threatening Gambol and Rachel, in order to seek some twisted personal justification for wanting to kill them.