Here is the full list of all 53 The Mageseeker: A League of Legends Story achievements.
Pet the horse, Azucarillo
Hook 10 times in less than 8 seconds
Unlock all spells
Complete the game
Defeat Wisteria
End Hesbeth's experiments
Complete all secret lab raids
Defeat Nocturne
Stop the mage execution at Meltridge
Defeat Wisteria in Meltridge
Complete the fourth enemy wave
Unlock all ultimates
Get Morgana's vision
Complete the assault to the Great City of Demacia
Defeat Jarvan and Shyvana
Defeat Garen
Complete the third enemy wave
Complete Cloven Dell Headquarters assault
Explore the Central Headquarters
Defeat Killan
Unlock all slots improvements
Complete the second enemy wave
Visit Lux in Terbisia
Defeat the Rattleclaw
Complete Quickmoore Headquarters assault
Pet the dog, Sam
Destroy the Capelworth Headquarters
Complete Iffighain Headquarters assault
Free the mages imprisoned in Glyndemere Headquarters
Fight Rayn in Glyndemere
Complete the first enemy wave
Unlock the Unchained Mode
Meet Morgana
Fight Morgana
Copy abilities from enemies 200 times
Rescue Kara from the Mageseekers
Defeat Wisteria in Devineur
Defeat Rayn in Devineur
Hook and whirl 100 times
Complete Cypress Glen Headquarters assault
Use opposite magics 150 times
Rescue Yops from one of Hesbeth's labs
Defeat the Giant's Helm
Complete Ternmoat Headquarters assault
Recruit Rukko and find information about Kara's whereabouts
Unlock a spell from Leilani
Explore the Dark Forest
Defeat the Polar Lizard
Kill 5 enemies at the same time
Escape from the Great City of Demacia
Fight Garen at the beginning of the game
Find all Silverwings
Unlock all achievements