Here is the full list of all 10 10 Minute Barbarian achievements.
You made it through the intro.
Victorious in the Northern Wastes
Own 10 Trees
Got 4 dragons
Killed more than 1200 enemy in one mission
Got 1200 of your own troops killed in one mission
Won a Queen's Guard award for an amazing score
Hired a Necromancer
Saved the world in your own particular...idiom
Beat every mission as Lord High Emperor