For this you have to kill all the non zom-bears, meaning a zom bear cant kill a bear.
For one it doesnt matter if a bear kills a zom bear, you will still get the achievemnt if you do it right, and the achievement doesnt pop up as soon as you have killed all bears. After youve completed and your scores are tallyed up you will get the achievement.
My best advice would be to use someone fast, like Master Naughty and run and find all normal bears and kill them, and then kill the rest and progress to the end.
Good Luck :)