Here is the full list of all 31 Realm of the Mad God achievements.
Unlocked Rogue Class
First in a long line of heroes
Unlocked Archer Class
Unlocked Priest Class
Unlocked Warrior Class
Unlocked Knight Class
Unlocked Paladin Class
Unlocked Assassin Class
Unlocked Necromancer Class
Unlocked Huntress Class
Unlocked Mystic Class
Unlocked Trickster Class
Unlocked Sorcerer Class
Reached Level 10
Reached Level 20
Achieved fame of 500
Achieved fame of 1000 or more
Completed over 13 Class Quests
Completed over 26 Class Quests
Completed over 39 Class Quests
Completed over 52 Class Quests
Completed over 65 Class Quests
Completed every dungeon type with a single character
More than 1 million tiles uncovered with a single character
More than 4 million tiles uncovered with a single character
More than 100 party member level ups with a single character
More than 1000 party member level ups with a single character
More than 1000 quests completed with a single character
Defeated more than 100 gods with a single character
Defeated more than 1000 gods with a single character
Dealt Killing blow to Oryx