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