Here is the full list of all 37 The Last King's Archer achievements.
Collect 70000
Complete Mushroom Infestation Mode
Collect 50000
Complete Globin's Party Mode
Protect The King
Kill 50 Mushroom Creature
Shoot 30 Thunder Arrows
Complete All Thunder Dragon's Upgrades
Collect 30000
Kill 1000 Enemies
Kill 100 Flying Creature
Kill 500 Globins
Shoot 50 Fire Arrows
Complete All Fire Dragon's Upgrades
Kill 50 Flying Creature
Shoot 30 Fire Arrows
Shoot 10 Thunder Arrows
Kill 300 Globins
Complete All Archer's Upgrades
Collect 10000
Kill 10 Flying Creature
Shoot 1000 arrows
Shoot 10 Fire Arrows
Collect 5000
Shoot 5 Thunder Arrows
Kill 50 Bigger Globin
Kill 100 Globins
Shoot 500 arrows
Die 10 times
Kill 10 Bigger Globin
Shoot 100 arrows
The King is Dead
Die
Collect 500
Kill 10 Globins
Shoot 10 arrows
Complete Flying Attack Mode