Chosen by The Neverborn, i.e. the primordials who "died" in the original war. The thing is, primordials predate the literal existence of death, so they can't die. Instead they're trapped in endless suffering which can only end if they're no longer anchored to Creation—that is, if Creation is destroyed.
This horrible death-flavored paradox has ripped a hole in the fabric of existence. The hole is called Oblivion, and the task of Abyssal exalts is (ostensibly; they can go rogue) to cast all of Creation into said Oblivion.
Abyssal exalts are dark mirrors of Solars; they're all basically the same as solar castes but death-flavored. Unlike Solars, who exalt when the Unconquered Sun sees them about to attempt the impossible, Abyssals............. did not get noticed. Rip. They had the potential, but there wasn't a Solar exaltation waiting for them, so instead they [almost] die. On the brink of death, they are then given the opportunity to come back as an Abyssal, in service to the Neverborn.
Most but not all Abyssals serve under a Deathlord. Deathlords were the first recruits the Neverborn made—and they were recruited from the superpowered pissed off ghosts of First Age solar exalts who died in the usurpation. Deathlords have their own personalities and flaws, but in THEORY they're 100% obligated to serve Oblivion.
(It is OOCly but distinctly not ICly known that Abyssals who really truly try to redeem themselves can turn into Solars.)