Here is the full list of all 19 Idle Chess Story achievements.
Win 10 Games.
Win 500 rounds
Deploy 500 Structures to fight.
Deploy 500 Undeads to fight.
Deploy 500 Guardians to fight.
Win 300 Rounds
Deploy 500 Warriors to fight.
Deploy 500 Elementals to fight.
Deploy 500 Wizards to fight.
Deploy 500 beasts to fight.
Win 150 Rounds.
Win 75 Rounds.
Win 25 Rounds.
Win 1 Game.
Kill the Engineer
Kill the Alien Soldier
Win 10 Rounds.
Kill the Eyebat
Earn 10000 Gold.