Here is a list of the Dark Souls 3 Achievements names and the task to complete them. There are a total of 43.
Enkindle – Light a bonfire flame for the first time.
Embrace the Flame – Become a Host of Embers for the first time.
Iudex Gundyr – Defeat “Heir of Fire” Iudex Gundyr
Vordt of the Boreal Valley – Defeat Vordt of the Boreal Valley
Covenant: Way of Blue – Discover Way of Blue covenant.
Curse-rotted Greatwood – Defeat the Curse-rotted Greatwood.
Covenant: Blue Sentinels – Discover the Blue Sentinels covenant.
Crystal Sage – Defeat Crystal Sage.
Lords of Cinder: Abyss Watchers – Defeat the Abyss Watchers, Lords of Cinder.
Covenant: Warrior of Sunlight – Discover Warrior of Sunlight covenant.
Deacons of the Deep – Defeat the Deacons of the Deep.
High Lord Wolnir – Defeat High Lord Wolnir.
Pontiff Sulyvahn – Defeat Pontiff Sulyvahn.
Covenant: Watchdogs of Farron – Discover Watchdogs of Farron covenant.
Lord of Cinder: Yhorm the Giant – Defeat Yhorm the Giant, Lord of Cinder.
Lord of Cinder: Aldrich, Devourer of Gods – Defeat Aldritch, Devourer of Gods, Lord of Cinder.
Dancer of the Boreal Valley – Defeat Dancer of the Boreal Valley.
Old Demon King – Defeat Old Demon King.
Dragonslayer Armour – Defeat Dragonslayer Armour.
Covenant: Rosaria’s Fingers – Discover Rosaria’s Fingers covenant.
Oceiros, the Consumed King – Defeat Oceiros, the Consumed King.
Untended Graves – Reach the Untended Graves.
Champion Gundyr – Defeat Champion Gundyr.
Lord of Cinder: Lothric, Younger Prince – Defeat Lothric, Younger Prince, Lord of Cinder.
Archdragon Peak – Reach Archdragon Peak.
Supreme Weapon Reinforcement – Reinforce any weapon to the highest level.
Ancient Wyvern – Defeat Ancient Wyvern.
Covenant: Mound-makers – Discover Mound-makers covenant.
The Nameless King – Defeat Nameless King.
Covenant: Aldrich Faithful – Discover Aldritch Faithful covenant.
Covenant: Blade of the Darkmoon – Discover Blade of the Darkmoon covenant.
Ultimate Estus – Reinforce the Estus Flask to the highest level.
Ultimate Bonfire – Reinforce a bonfire to the highest level.
To Link the First Flame – Reach “To Link the First Flame” ending.
The End of Fire – Reach “The End of Fire” ending.
The Usurpation of Fire – Reach “The Usurpation of Fire” ending.
Master of Infusion – Perform all forms of infusion.
Master of Sorceries – Acquire all sorceries.
Master of Pyromancies – Acquire all pyromancies.
Master of Expression – Learn all gestures.
Master of Miracles – Acquire all miracles.
Master of Rings – Acquire all rings.
The Dark Soul – Complete all achievements.
If there are any missing please add a note about that in the comments below.