Here is the full list of all 35 Soul King achievements.
You started a new game
Rescued your first Dreamer
Reached the dev room
Reached The Library
Reached the Barclay Reserve Caverns
Dethroned the Soul King
Finished Soul King. Hope you enjoyed it!
Reached Eileen's Cabin
You defeated the Soul King's most powerful grunts
Defeated the Soul King's favorite goon.
You found Grimzo's lab hidden in the Caverns
Found all of Mochi's napping spots.
Destroyed 50 Soul Fiends
Destroyed 50 Soul Scouts
Destroyed 10 Soul Snipers
Destroyed 10 Soul Knights
Destroyed 10 Soul Turrets
Died 10 times
Caught by the wall 5 times
Drank 10 Armor Teas
Drank 10 Stamina Teas
Drank 10 Health Teas
Drank 10 Mello Teas
Fired 100 rounds from any Blaster
Fired 250 rounds from any Blaster
Executed 15 melee attacks
Executed 15 Tool uses
Fired 500 rounds from any blaster
Taunted the first time
Got +3 multiplier by taunting downed enemies
Got +7 multiplier by taunting downed enemies
Got +10 multiplier by taunting downed enemies
Survived death by catching a Vitali-Tea dropped by an enemy as you fell
Customized your Hearticon in the Reserve
Finished your first Gauntlet run