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