Like all the other Endurance Achievements, first choose the challenge with the same name.
This is best attempted at the lowest difficulty, Adventurer. Critically injured is considered when your screen changes to black and white.
Find a group of Trinity soldiers and silently take out all but one. Then have one player slip around the side and hide while the other runs up to him. Wait until the player getting attacked signals they are critical, then kill him for the Achievement to unlock.