While on Balboa Island you will find the Mama Bear which is a bear much larger than the other bears found on the island. Not only is she larger but she also has much more health and does much more damage. It is not required that you fight her unless you want the achievement.
Blueberries can be found on bushes all across the island. These can then be placed into the red container with the blue tarp near where she spawns in order to distract her.
The Mama Bear is dangerous to engage with melee weapons, even with two people, so I recommend using a bow. Near the Mama Bear's area, to the left of the cave entrance, is a couple of boulders which you can climb in order to get onto a ledge. This ledge is a safe spot that the Mama Bear cannot hit you on. From this ledge, you can shoot the Mama Bear with a bow and in some cases she wont even aggro you.
It will take between 40 - 50 arrows to kill the Mama Bear so make sure you come prepared. Make sure to grab the machete from the cave near Mama Bear if you haven't already because it will be a pain to come back to this area to get it once you are across the island.