Here is the full list of all 20 Beastrun achievements.
level a relic up to level 5
level a skill up to level V
level a unit up to level 10
find the smelly hat
execute 30 skills in the first 5s of battle
clear the final room in a dungeon
Deal 40 damage or more in a single hit
defeat an opponent with combined unit levels that are at least 10 higher than yours
level a skill up to level IV
reach the meadows
reach the forest
win a battle that lasted at least 90s
level a skill up to level III
Heal 100 health in one battle
reach the fields
deal 50 combined fire damage in one battle
get a super rare unit
level a unit up to level 5
level a unit up to level 2
win a battle with 25% of enemy skills failed