Here is the full list of all 15 Heroes of Havoc: Idle Adventures achievements.
Unlocks the ability to upgrade your heroes' rarity
Unlock Timed 3 - The Green Meanies
Unlock Trial 3 - Only Human
Unlocks the mysterious power of Relics
Unlock Timed 2 - Dungeons and Dragons
Unlock Trial 2 - Rise of the Robots
Unlock Idle Mode - Heroes will continue fighting without you
Unlocks the ability to upgrade your equipment
Unlock Crusade Mode
Unlock the Visting Merchant
Unlock the Chest in the Main Menu
Unlock the competitive PVP modes by opening the Dark Portal
Unlock Elite Campaign
Unlock Timed 1 - The Hunted
Unlock Trial 1 - The Impvasion