There are many references regarding how his scars were obtained. But none of the ones I know about hold any true origin. I am an avid fan of the Joker and for the last few years I have been searching for the source that describes the origins of his scars most clearly and faithfully.
In the batman movie from 1989 directed by Tim Burton, he get some chemicals in his face and then falls in a giant "pot" of other chemicals and escapes. He then go to a "underground" surgeon who opperates on him. You don't know exactly what he does to the joker other than it is his face he operates on. My guess is that the chemicals that the joker got in his face destroyed his cheeks and then then the surgeon managed to put the cheeks back together. The scars is therefor a result of the cheeks first destroyed and then repaired.
Another theory could be that the chemicals made the Joker mad and in his madness the joker told the surgeon to carve scars into his cheeks to make him more "happy"
In the original batman by tim burton the joker is based on the original writings of the dc comics and when he is in the chemical factory from been set up by the mob he's involved with because he's having an affair with the mob bosses wife, when batman comfronts him, he shoots and batman reflects it and it goes through his cheeks and he falls Into a container of chemicals that turn his skin white and his scars bright red
The joker got his child abuser (his mom) and she took a knife and cut the scars to make him look happy for her but he never was happy with her around so the dad followed with the mom and dad wanted to beat him up so his dad turned into batman and he knew but joker didn't so they kept it a secret and when joker found out he became the viscous villain Joker
His father was abusive, and one night, after killing his mother, his father took a shard from the beer bottle he was drinking and stabbed the Joker in the stomach. The air he breathed out was short and suppressed, so it sounded like he was laughing. His dad was furious and took the blade and carved it across his face. He then took the Joker's mom's favorite lipstick and smeared it on the wound, making it incredibly painful. The Joker then bled internally until a woman named Joyce helped him after hearing his cries. She was the wife of the head of the mob, who took him in. Years later, they asked him to kill Bruce Wayne, because he had murdered one of their mobsters. He was unsuccessful, so they pushed him into a pit of bleach. He then was taken out only to be assured that if he ever killed Batman, he would be welcomed. But, until then, the mob would send his name out to bounty hunters, wanted dead or alive,