The 'Better Luck Next Time' achievement is one that doesn't have a set trigger to unlock and is completely random on when it could happen.
For an enemy to cheat death, they need to be of the captain class. When you kill them, sometimes they will seem dead in the nemesis menu and drop a reward like other captains but will actually have a small chance and remaining alive, even after your first fight. You will have the chance to run into them randomly or through the nemesis system where they will most of the time become stronger then when you previously fought. After you meet each-other again the achievement will unlock instantly.
Here is a video showcasing how such an encounter works.