Here is the full list of all 14 Bearslayer achievements.
Finish any level without dying.
Kill two enemies in one level.
Kill the bear.
Kill 10 enemies.
Kill 3 ghosts.
Jump on jump pads 50 times.
Finish any level three times.
Reach 1000 deaths
Finish any two levels without dying.
Take one million footsteps without dying.
Make 500 wall jumps.
Finish any secret level without dying.
Kill the Witch boss.
Kill 3 imps.