Here is the full list of all 37 The Slimeking's Tower achievements.
The Collector Unlocked
Jade Unlocked
Reach The Portal
Attack a mimic before it's transformed
Reach Sigma
Reach Theta
Reach Tau
Reach Ohm
Reach Pi
Maiyan Unlocked
Reach the basement
Reach Omega
Reach Gamma
Midas Unlocked
Igneous Unlocked
Reach Epsilon
Reach Delta
Defeat Lodeslime
Reach Beta
Get your first boss key
Play Adventure Mode for the first time
Unlock Hairry's secret item
Beat the portal
Dots Unlocked
Mind Unlocked
Void Unlocked
Prayer Unlocked
Dots Completed
Mind Completed
Maiyan Completed
Midas Completed
Igneous Completed
Void Completed
Jade Completed
The Collector Completed
Prayer Completed
Hairry Completed