Here is the full list of all 19 Hextech Mayhem: A League of Legends Story™ achievements.
Beat any level in Impossible Mode
Get 100% Bonus Mayhem in a level
Earn Grandmaster Rank in any level in Full Action Mode
Beat Boss 3
Collect 72 Cyan Gizmos
Beat Boss 2
Earn a Master Rank in any level in episodes one through five
Collect 48 Cyan Gizmos
Beat Boss 1
Earn a Diamond Rank in any level
Collect 24 Cyan Gizmos
Get 100% Action Prompt Accuracy in a level
Beat a level without bonking
Earn a Grandmaster Rank in any level in episodes six through ten
Earn a Challenger Rank in any level in episodes eleven through fifteen
Earn a Challenger Rank in every level in Normal Mode
Earn a Challenger Rank in every level in Full Action Mode
Earn a Challenger Rank in every level in Impossible Mode
Collect all Gizmos in the game in Normal Mode