A villain should be the opposite of his nemesis, the yin to their yang, while that is exactly what The Joker is to Batman. The Joker is the chaotic, playful, murderer that Batman is not. Like with the Green Goblin and Spider-Man this single villain has shaped his rivals world in ways no other villain has. From paralyzing Batgirl with a bullet to the spine, or killing Robin II with a crowbar he has been the true arch enemy to the Dark Knight.
I recommend reading ‘The Killing Joke’, ‘The Long Halloween’ and ‘Death in the Family’