Here is the full list of all 13 Exodemic achievements.
Win a run with the Spartan ship
Win a run with the Ghost ship
Win a run with the Behemoth ship
Win a run on Normal difficulty
Win a run on Hard difficulty
Win a run without taking any damage
Win a run without letting any enemies escape
Win a run without visiting any medics
Win a run with less than 30 accrued contagion
Win a run without equipping any ship augments
Win a run without equipping any dash augments
Win a run without equipping any weapon augments
Win a daily challenge run