While playing as the Lost, the player notably begins with flight and without any health, and as such will die as a result of any damage taken. Note that resurrecting with the Ankh, Lazarus' Rags or Judas' Shadow and dying in the appropriate fashion still fails the requirements and you will have to start over.

Azazel must kill the fallen then die from Satan's second or third stage.

Being killed by anything else in the fight will cause the run to fail and you will need to start over from the beginning.

Judas must die by Mom's foot or Mom's hand during the boss fight.

Magdalene must die from bombing herself in the Caves/Catacombs (can be XVI The Tower tarot card).

Isaac must die from a Mulliboom (Muliiboom spawns in the door under you in this seed JTMD EZPF).

The Lost is unlocked via very specific deaths which must be done in order without closing the game or dying erroneously between runs. The Lost is one of the 11 Playable Characters in The Binding of Isaac: Rebirth Starting Activated and/or Collectible Items:

The lost binding of isaac