To unlock this, you have to finish 3 victory laps and start your 4th.
To start a victory lap, you have to go to the Dark Room and defeat The Lamb. A pop up will immediately appear and ask if you want to do a victory lap. Choose yes, and you will go back to the beginning of the game with MOST of your items. (a couple may be removed but it seems kinda random as to which ones) On every lap, you MUST fight The Lamb again, as any other boss doesn't give you the victory lap option.
On the first and second victory laps, you will take 1 full heart of damage on any hit on every floor unless you have the Wafer. Finishing the first victory lap will get you
On the third lap, your character becomes The Lost, but won't start with the D4 or Holy Mantle unless you picked them up in a previous lap. Also, shops no longer appear. When you beat The Lamb on this lap and choose to start the next victory lap, the achievement unlocks.