The Joker's originas
Two versions of it that are most popular, one being that he was a failed comedian with a pregnant wife, he is desperate for money and joins a gang that runs a villain called the Red Hood, by each switching off as the Hood when they go about their crimes. While trying to rob a place, The Joker is grief stricken and takes on the heist. He falls into a vat of toxic chemicals while Batman tries to thwart him and his hair is bleached and so is his skin. That's The Killing Joke version.
The second being that he was he was a bored hitman and started committing violence at random and meets Batman for the first time, the rush the meeting creates prompts him to do more heinous killings and even capture Bruce Wayne's love at the time. Batman is furious so he sends the mob to kill the guy, and the mob takes him to an abandoned facility, he falls into a vat and punctures a tank above him. Batman throws a batarang at him and he escapes but the batarang creeates a glasgow grin for the Joker.
I like you version, but it's doesn't catch his insanity, his sorrow or his good taste in clothes.