Killing the Breeder in less than 5 minutes can be prove to be difficult on your first go around, but isn't too bad if all 4 players know their roles and what to do.
First of all, it is extremely important for all 4 players to have the Venom-X
weapon when entering the Final Area. It is recommended that all players save up as much ammo as possible for this weapon. Also, make sure that all the players share cash, so that everyone has a good amount to help each other out with ammo, shields and boosters. Money is hard to come by in the final area because the enemies are rare, and don't give much cash.
As soon as the first player enters the broken wall and the audio dialogue begins, the timer will begin. The tactic is simple, players take turns using the Venom-X to shoot the Breeder in the head, causing the green toxic gas to stay on it. Once the monster takes damage, it will fall and kneel, leaving it 100% vulnerable to direct attacks from up close. Do not use Venom-X during this stage. You want 3 players to be unloading their fully automatic weapons with explosive ammo into its weak spot. Having one member with a fully upgraded Death Machine will work wonders, but is not necessary. The other member will keep smaller enemies that spawn away from the group.
As soon as the Breeder begins to come out of the kneeling position, RUN! You want to seek cover behind one of the hives or work trucks in the area. Watch out for its machine guns, they are extremely powerful. If the Breeder launches eggs, destroy them before they get a chance to hatch, and dispose of small enemies when possible. The Breeder also has a "shielding" period which I believe it invulnerable to Venom-X and regenerates some health. While players are covering the Breeder with Venom-X ammo, the other players will want to aim at the Breeder's head or watch out for small enemies. Rinse and repeat the cycle until it gets down to 50% health, hopefully in about 2 minutes. At this point it will go hide, and a ton of enemies will spawn. Use the small gap in time to regenerate ammo and shield with the team. While she is gone, a ton of difficult enemies will spawn including 3-4 Rhinos. If money/ammo permits, it is recommended that the team uses some sentries/vultures as well extra Venom-X ammo on the Rhinos to bring them down a lot faster and protect the team better. Once the Rhinos are cleared, you want to be under the 3 minute mark with a decent amount of money and Venom-X ammo. Use the above method to kill the Breeder, by using Venom-X ammo and headshots to make the Breeder fall, and then have the entire team shoot her in the weak area. When she is close to dying, use as many abilities as you need to stay alive, and aim for the head to bring her down. Once her health bar disappears, she is dead and the timer stops.
Hopefully your team did it within the 5 minutes, and the achievement will unlock for everyone! It is not recommended to try and get this achievement while using a relic, although it is possible.