Currently a glitched achievement and is unobtainable.
To kill a bear, it's more expensive on resources than it is hard.
3 spear traps are enough to kill a bear at full health.
There are several occasions at nighttime where wolves will get in a battle with the bear. Unfortunately even after a couple wolf swipes on the bear, it STILL takes 3 spear traps to kill it. Stay decently close to the raft area until the battle is over because i've once had the bear aggro on to me after he was done with the wolves and I almost died.
Line up the 3 spear traps in a line like you're making a track for the bear. Give just a couple feet in between each trap. Shoot an arrow to aggro the bear from the cave.
When a spear trap springs, it staggers the bear. It will soon release itself and head for the player again. If you've done it correctly it should just run into all 3 and kill itself on the third one.