Here is the full list of all 29 Battle Sweeper achievements.
Reach level 20 in Classic Mode
Reach level 15 in Classic Mode
Reach level 10 in Classic Mode
Reach level 5 in Classic Mode
Reach level 20 with the Rogue
Reach level 15 with the Rogue
Reach level 10 with the Rogue
Reach level 5 with the Rogue
Reach level 20 with the Wizard
Reach level 15 with the Wizard
Reach level 10 with the Wizard
Reach level 5 with the Wizard
Reach level 20 with the Lizard
Reach level 15 with the Lizard
Drink 10 Potions
Reach level 10 with the Lizard
Reach level 5 with the Lizard
Drink 5 Potions
Kill 20 monsters
Kill 15 monsters
Reach level 20 with the Knight
Kill 10 monsters
Reach level 15 with the Knight
Kill 5 monsters
Drink a Potion
Reach level 10 with the Knight
Kill a monster
Reach level 5 with the Knight
Finish a level without fighting any monsters