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