There are four wanted lists in the game. You must defeat all five monsters on all four lists.
Remember, wanted list monsters only appear at night.
The lists are as follows:
Mika's Monster Notebook
Location: Near the Moon Cave Entrance
Demon Locations: Shinshu Field
Demon Type: Red Imp
Haruka's Revenge Contract
Location: Kusa Village's Inn
Demon Locations: Taka Pass
Demon Type: Dead Fish
Masu's Monster Manifest
Location: Yama's Restaurant in Sei'an City (Night only)
Demon Locations: Ryoshima Coast
Demon Type: Thunder Ear
Wali's Record of Penance
Location: Wali's hut in Kamui
Demon Locations: Kamui
Demon Type: Igloo Turtle
When you've defeated the 20th, the achievement will pop. To make sure that it doesn't glitch, make sure you save before hunting for the 20th enemy.