Here is the full list of all 53 FateEXTELLA achievements.
Performed a Moon Crux/Drive for the first time.
Equipped an Install Skill for the first ime.
Equipped a new Mystic Code.
Cleared the 1st stage of the Flame Poem Arc.
Revealed the true name of your Noble Phantasm for the first time.
Switched with a vice general for the first time.
Performed an Extella Maneuver for the first time.
Obtained all achievements.
Defeated Tamamo no Mae in the Flame Poem arc.
Cleared Artoria Pendragon's hidden side mission, then defeated her, once.
Defeated Altera in the Flame Poem arc.
Cleared the Flame Poem arc.
Defeated Nero Claudius in the Orchid Poem arc.
Defeated Altera in the Orchid Poem arc.
Cleared the Orchid Poem arc.
Defeated Tamamo no Mae in the Dawn arc.
Cleared the Dawn arc.
Cleared Artoria Pendragon's hidden side mission, then defeated her, on four different stages.
Allied with Tamamo no Mae in the Golden Poem arc.
Cleared the Golden Poem arc.
Completely cleared Cú Chulainn's sidestory.
Completely cleared Medusa's sidestory.
Completely cleared Lü Bu's sidestory.
Completely cleared Karna's sidestory.
Completely cleared Elizabeth Bathory's sidestory.
Completely cleared Jeanne D'Arc's sidestory.
Completely cleared Iskandar's sidestory.
Completely cleared Gilgamesh's sidestory.
Completely cleared Artoria Pendragon's sidestory.
Completely cleared Li Shuwen's sidestory.
Completely cleared Gawain's sidestory.
Completely cleared Nameless' sidestory.
Completely cleared Archimedes' sidestory.
Cleared all Servants' sidestories.
Performed 200 Extella Maneuvers.
Performed 50 Moon Crux/Drives.
Revealed the true name of your Noble Phantasm 20 times.
Recorded all Install Skills in the gallery.
Upgraded one Install Skill by +10.
Upgraded one Install Skill by +20.
Prevented a Sector Trap for the first time.
Prevented a Sector Trap 30 times.
Recorded all Mystic Codes in the gallery.
Succeeded 25% completion rate in the gallery.
Succeeded 100% completion rate in the gallery.
Obtained an EX rank in the results.
Defeated a total of 1,000 Aggressors.
Defeated a total of 5,000 Aggressors.
Cleared all stages on VERY HARD difficulty.
Conquered a total of 100 sectors.
Raised a Servant's Kizuna level to 30.
Raised 3 Servant's Kizuna level to 30.