Here is the full list of all 13 A Healer Only Lives Twice achievements.
Unlock all upgrades
Complete TRIAL Mode
Attack three enemies at one time by using Courage skill
Complete EX MAIN Mode
Complete MAIN Mode or EX MAIN Mode
Learn all skills
Defeat Troll Thug
Defeat Gryphon
Defeat Minotaur
Defeat Golem
Game over
Enter Tier 3 (MAIN Mode or EX MAIN Mode)
Cure Petrification or Poison by using items or skills