Well, this quest should be pretty easy to figure out. Just kill the right amount of bears that are listed. However, this it not listed in the mission panel (scroll icon), it is listed in your journal (book icon at bottom right).
Try your best to stay away from frezied bears unless you have potions, these are not on your list of bears to kill.
If you start a train of bears, if you are far enough, you can get to the door after ten (10) seconds of being attacked, if not, you will have to fight them.