To save some time, play on LOW CHAOS first. The game automatically makes permanent saves for each chapter of the DLC that you have to manually delete yourself. At the end of your LOW CHAOS run and you're deleting old unnecessary saves to save room on your quickly shrinking hard drive, keep the EMINENT DOMAIN auto save. This will start you on the 2nd chapter.
When you are ready for the High Chaos run, load the Eminent Domain save and kill everyone, get spotted, make noise (use your gun, grenades, explosive bolts, etc.), feed the rats delicious bodies. You only need to get High Chaos in 2 of the 3 chapters for the achievement, eliminating the need to play the first chapter a 2nd time.
This is the fastest way to get the achievement.