You'll have to survive 18 bear attacks. It seems only Grizzlys will attack you. American black bears run away.
When you find one, select your repeating rifle (fast reload and good amount of dmg), but don't shoot it yet.
Let the bear mangle you down and spam the B button. You will break free eventually and have the chance to shoot at the bear. Quickly press RT (three to) four times. That should kill the bear while you're still on your back on the ground and give you progress towards the achievement.
If you upgrade your weapon and/or use better bullets, you'll kill the bear faster. You save some ammo and you have a better chance to make it count.
Grizzly bears can be found in the region Grizzlies East in Ambarino. There are muliple spawns north east of the legendary bear (north and south of the railroad tracks). For exact locations, see the picture down below. You can sell the hide to the trapper and return later or save the game and reload or camp. Leaving the area (e.g. visiting the trapper, works best).
I used the one near Elysian Pool. This has the chance to be far enough to reset at least one spawn. While reaching the spawn south of the railroad, the bear was not there most of the time. However, it should be enough to cross the tracks and then turn around. This can force a bear-spawn, but that is not 100% guaranteed.
Location and best way to scout both locations
Maka made a great video explaining the achivement and a different approach. Try it, it might work better for you too!