Here is the full list of all 61 Hero's Hour achievements.
Win a game as Enclave (normal or harder)
Win a game as Earthen (normal or harder)
Win a game as Tide (normal or harder)
Win a game as Wild (normal or harder)
Win a game as Horde (normal or harder)
Win a game as Lament (normal or harder)
Win a game as Pyre (normal or harder)
Win a game as Decay (normal or harder)
Win a Challenging difficulty game
Win a Hardcore difficulty game
Win a game as Arcane (normal or harder)
Win a 5+ player game
Win a 7-player game
Win a game on the Patience map in less than 24 real-time hours
Put 200 power worth of units to work at a Slave Pens
Win a game without casting magic
Win a game without any artifacts
Win a game as Pillar (normal or harder)
Learn an adventure map spell at an Archmage Tribunal
Win a game in less than 15 minutes
Win a game in less than 30 minutes
Win a Normal difficulty game
Win a game as Order (normal or harder)
Have a hero reach level 30
Win a game before week 3
Win a game without hiring a hero
Win an Easy difficulty game
Have a hero reach level 20
Have a hero with an artifact in all equipment slots
Achieve rank 5 in a hero skill
Have 100000 gold
Have three artifacts in a set
Acquire a treasure elemental
Have a hero with a sum of attack, defence, knowledge and spellpower above 100
Defeat a near impossible enemy army
Defeat a challenging enemy in battle using only elementals
Complete a full artifact set
Win a battle without taking casualties
Recruit 666 Pyre units in one game
Defeat the red dragon in Skirmish mode
Defeat the ghost larvae in Skirmish mode
Defeat the black dragons in Skirmish mode
Acquire a dragon via Diplomacy
Fully develop a town
Save 500 creatures using infirmaries and hospitals in a single game
Devour a Black Dragon
Create an Idol
Convert a town to a faction you did not start as
Use a Forlorn Cloister with your highest levelled hero
Bloodwarp a Hatchling into a Seraph
Enter combat against an enemy player with at least 2000 power worth of army
Control 10 overgrown mines
Enter combat with a hero that uses the most overpowered Rune
Cast Shadowcasting while your hero has 50 or more spellpower
Win a game on Hardcore+ and with hamlet starting conditions
Find an obelisk treasure with just 1 clue
Encounter 95% of all rare creatures
Encounter 95% of all artifacts
Encounter 95% of all spells
Encounter 95% of all adventure buildings
Achieve 95% of all achievements