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