Here is the full list of all 44 Fate/EXTELLA LINK achievements.
Cleared all Extra Stages.
Defeated two bosses in the "EX: Facsimile Fortress Nightmare" stage.
Cleared 32 Extra Strages.
Combined two combined Install Skills.
Cleared 24 Extra Stages.
Unlock 75% of the Gallery.
Cleared 16 Extra Stages.
Raised three Servants' Bond levels to MAX.
Cleared eight Extra Stages.
Defeated Karna as Arjuna.
Defeat Artoria as Lancelot.
Combined two Install Skills.
Used Karna's "Oh Brahma, Cover the Earth" or "Oh Brahma, Split the Earth" skills.
Defeated Scathach as Cu Chulainn.
Defeated Iskandar as Darius III.
Cleared one Extra Stage.
Cleared Stage 27 and viewed the ending.
Raised a Servant's Bond level to MAX.
Unlock 50% of the Gallery.
Chose to listen to Archimedes.
Cleared Stage 19 and viewed the ending.
Cleared stage 13 or 14.
Cleared Stage 11.
Cleared Stage 18 and viewed the ending.
Cleared Stage 15 or 16.
Cleared Stage 9 or 10.
Maxed out an Install Skill.
Earned a Mystic Code for strengthening a bond or clearing a stage.
Earned three Mystic Codes for strengthening bonds or clearing stages.
Unlock 25% of the Gallery.
Cleared Day 3.
Executed 8 Rush Attacks in one stage.
Cleared Day 2.
Crafted a Mystic Code.
Cleared Day 1.
Used a Noble Phantasm twice in one stage.
Activated Moon Drive four times in one stage.
Viewed the event in which Charlemagne first appears.
Gained 1000 KO's in one stage.
Acquired all other Trophies.
Cleared Stages 1 through 27 on Very Hard.
Unlock 100% of the Gallery.
Unlock 100% of the Mystic Codes.
Unlock 100% of the Install Skills.