Mostly luck. Just try to stay back. If you're support, use the cloak when the monster goes after you or put up your shield if you're assault.
I suggest setting up a solo custom match to your preferred monster hunting map (I like the Dam, it's a very simple map) on Hunt with a timer of five minutes. I played as the assault Parnell, with the rest of the starting cast. Daisy is useful in helping track the monster who will try to trick you with fake paths, Hank and Val will help keep the team alive. Set the monster as Goliath, moves are easy to dodge and it has less mobility than the other two. As Parnell never use his LB equipment which hurts you, I believe losing any health period will disable the achievement for that match.
Now track down the monster and once you find him keep your distance and fire away with the rocket launcher. Stay up high and if he starts to get near your area, try to move to another high spot or just be ready to activate your shield with RB. With the timer so low, once your trappers dome runs out the monster will continue to fight you so it won't lose.
If you take any damage at all, just press start and restart the match.
I believe the hunter team must be the ones that kills it. I went my first match with taking no damage but it died to wildlife.