Here is the full list of all 23 Monstro: Battle Tactics achievements.
Complete Introduction campaign
Introduction #8 - Kill all the goos
Introduction #3 - Make the gargoyle step on each web trap and complete the mission
Complete Human campaign
Complete Human++ campaign
Complete Monster campaign
Complete Monster++ campaign
Read credits
Introduction #6 - Kill the minotaur with the spike trap
Human #3 - Finish the mission without moving the mobile walls
Human #13 - Finish the mission without losing HP
Human #20 - Kill the slime and win the mission
Monster #7 - Finish the mission without losing a single unit
Monster #19 - Finish the mission without losing a single unit
Monster #20 - Complete the mission without losing a single unit
Human++ #8 - Complete the mission without losing a single unit
Human++ #12 - Complete the mission without losing a single unit
Human++ #15 - Kill a manticore and complete the mission
Human++ #19 - Complete without getting hit
Monster++ #4 - Finish the mission without losing a single unit
Monster++ #5 - Complete the mission without the shroom attacking
Monster++ #16 - Kill all enemies and complete the mission
Monster++ #19 - Complete the mission with 5 shrooms alive