The Joker is arguably one of the greatest villains in pop culture history. The complete antithesis of Batman in almost every regard, many say he the hero's perfect adversary. His origins are shrouded in mystery, making it impossible to determine who he is and where he came from. This theory, however, could explain who he is, and why he is so obsessed with Batman.
Keep in mind, this is merely a fan theory.
The Joker, in reality, is Bruce Wayne's older brother. For story purposes, we'll call him Jack. Jack was born around 5-10 years before Bruce, and he seemed like a normal child at first. However, he started to show significant behavioral problems, such as being overly aggressive to other children and possibly killing animals. It could also be possible that Jack may have tried to harm Martha Wayne while pregnant with Bruce.
The Waynes decided to have Jack placed in a juvenile mental facility to try to get rid of him. Their influence in Gotham could've helped hide what they perceived as an embarrassment, and may have fabricated a story that Jack had died from an illness. Some years later, after Bruce is born, Thomas and Martha are killed.
Years later, Jack left the facility, either escaping or becoming of legal age, and supported himself through petty crime. He is constantly in a state of mixed emotions about his parents; anger at the fact that they threw him out, trying to yearn for what he would perceive as their approval despite their passing (similar to how Bruce uses his Batman persona) through his hard work and determination.
By now, Batman has appeared in the public eye. As Jack possesses an above average intelligence, it doesn't take him long to figure out who has the resources to gain access to high tech equipment, intense fight training and is based in Gotham. Deciding it's time for a "family reunion" Jack decides to adopt a costumed identity as the Red Hood. However, his meeting with Batman goes wrong, and is plunged into a chemical vat, dying his skin white, hair green, and lips red.
Jack takes the identity of the Joker and begins to continuely pester Batman. Joker doesn't care about material matters, nor the consequences of his actions (though he's aware of them). Essentially, Joker, like a brother, will continue doing what he can to cause Batman to retaliate to stop him. Very much like how one brother will annoy the other until they fight. The Joker doesn't want to kill Batman at all, he perceives it as picking fights with his brother. This is also why Batman won't kill him. Deep down, Batman knows him and the Joker are more alike than he wants to admit, something that prevents him from murdering him, but is unsure what.
Keep in mind, this is just a fun fan theory, not meant to be taken seriously.