Here is the full list of all 53 The Legend of Heroes: Trails of Cold Steel III achievements.
Completed the final chapter.
Used Burst in battle 50 times.
Completed Chapter 4.
Took all landscape photos and reported them to Vivi.
Forged a strong bond with Altina.
Forged a strong bond with Elise & Alfin.
Forged a strong bond with Ash.
Forged a strong bond with Alisa.
Forged a strong bond with Juna.
Won 600 battles.
Raised any two characters to Link Level 5.
Completed Chapter 3.
Raised a master quartz to level 7.
Used Rush in battle 100 times.
Used 100 S-Crafts in battle.
Completed Chapter 2.
Used over 500 follow-up attacks in battle.
Used Brave Order in battle over 100 times.
Inflicted Break over 800 times.
Completed Chapter 1.
Earned an S-Rank on any chapter.
Won 100 battles.
Won a battle with a tactical bonus of x3.0 or greater.
Completed the prologue.
Completed the orienteering exercise.
Mastered Trails of Cold Steel III and earned every achievement.
Achieved instructor rank S.
Achieved an academy grade of Prestigious.
Completed the game on nightmare difficulty.
Completed all quests, both at the branch campus and during field exercises.
Opened every treasure chest.
Completely filled the battle section in the notebook.
Completely filled the character section in the notebook.
Completely filled the books section in the notebook.
Completely filled the cards section in the notebook.
Completely filled the recipe section in the notebook.
Completely filled the fish section in the notebook. (All 25 types of fish + Rod Lv. Max)
Collected all radio material and reported it to Munk.
Gathered all 28 varieties of master quartz.
Acquired over a million mira.
Initiated 300 advantage encounters (including double and triple advantage).
Forged a strong bond with Kurt.
Forged a strong bond with Musse.
Forged a strong bond with Towa.
Forged a strong bond with Elliot.
Forged a strong bond with Laura.
Forged a strong bond with Machias.
Forged a strong bond with Emma.
Forged a strong bond with Jusis.
Forged a strong bond with Fie.
Forged a strong bond with Gaius.
Forged a strong bond with Millium.
Forged a strong bond with Sara.