Start a new run as the Bartender as he has the Charismatic trait which makes most people friendly. You'll need to find a 'Neutralize' mission; if there isn't one on your current level, then restart your run.
Once you've located your target, simply chat with him/her and ask them to leave town. You'll have a 50/50 chance of succeeding.
If you're successful, the target will leave and you'll earn the achievement.