One place I have always found a black bear is in the North East - Bou Reservoir, Open Hunt @ 13:00. There is a Drinking Area just to the NE of the starting location near the edge of the map (just a few steps away, you should get the noise indicator right away). There is a Presence hotspot between 13:00 and 15:00. Occasionally there is a few Caribou here with the bear but the bear has always shown up for me.
The location doesn't matter much, as long as you come across a BLACK bear and not a Grizzly.
To drop the bear you should have the 600-Grain Broadhead arrows equipped, these do more damage and penetrate deeper. I also equipped the scent cover so you can get in closer without spooking the bear. You can take additional arrows, but it is not necessary since you should be able to drop the bear in 6 shots.
You MUST be close enough to the bear to cause it to charge and attack you (40 yrds or closer), either sneak in close (my preferred way) or you will have to get lucky get the jump on it by using the terrain to get close enough without it noticing you.
Either way, you want to get within 40 yrds (preferably stealthy, I was about 35 yrds away when I got mine), any further out and it will more than likely run away. Once you get close enough, become unstealthy and get it to notice you, it should stand on two legs and growl at you.
Now, the fun part.
Shoot it in the face! No joke, this will really get it mad at you and encourage an attack. More than likely you will miss the brain, which is good... any vital hits will cause it to flee instead of attack you, that's why I say shoot it in the face.
Now, the hard part.
Fire arrows as quickly as you can into its face and chest, don't hold your breath just point and shoot. If you don't get a vital hit in those few shots, it should turn and run away... hopefully. (I did this several times and the second or third arrow always caused it to flee.) If it does flee, keep shooting it until it falls even if you are just shooting it in the butt. If you do get mauled to death, just repeat the steps and try staying a few yards further back before engaging the bear.
Once the bear drops, the achievement is yours.