The Batman who Laughs comes from Earth -22 in the Dark Multiverse. In this world, the Joker goaded Batman into killing him by committing various atrocities, such as bombing Gotham City, killing James Gordon, and infecting children with his Joker venom to make them like him. Unfortunately, after Batman killed Joker, the clown's body released one final, particularly powerful strain of Joker Toxin that infected Batman. In less than a week, the damage was done; Batman was turned into a new Joker, and slaughtered the Justice League and Batman Family before rampaging across his Earth, killing as he pleased.
After wiping out everyone on Earth -22 except for his "Robins", he was recruited by Barbatos to lead a team of evil Bruce Waynes called the Dark Knights.
He represents Batman's greatest, worst fear, namely that he could become like the Joker.