You can reach this ending by embracing Adam at the end of the Sanctuary. The achievement will unlock once the credits end.
You can get A Death in the Family if you immediately quit to dashboard once the credits start to play. If you did it quick enough, once you restart the game, you should have the option to continue from your last checkpoint, which will be on the bridge leading to Adam’s head. If the game autosaved, you’ll have to play through the game a second time.