Here is the tactics I used with a 4 player team.
Starting off, you can either buy the Kar98K or just pistol whip the freakbags. Fight until everyone has enough points to open doors and get one go at the mystery box. If you look directly at the Pack-a-Punch machine, there will be doors on your left and right sides. I used the doors to the Left of the Pack-a-Punch machine. Open doors along the way, making sure to divvy up the doors so everyone can keep as many points as possible.
Hang out at the box for as long as possible. At some point, the box is going to move anywhere on the map. Go to it only when necessary, making sure that everyone is back at the power generator ASAP. If you go to the box in the middle of the round, take someone with you. They can cover you while you grab your gun. By now, you should have enough points to open all the doors to the teleporters. You can do this with one zombie left in the round. Open all the doors and link the teleporters.
Eventually, your team will have to open the big door next to the generator so you can have a better fighting area, even if you didn't want to link the teleporters yet. Here is where the point factor will come in. By now, the entire team I was in had at least one Pack-a-Punch gun. Keep fighting on the catwalk like normal. The biggest thing with the catwalk: whoever is watching the window on top of the catwalk, needs to save one zombie. This person has to kill all but one zombie that comes to the window. When this zombie is last, one person can just stand at the window holding (X) to repair the window constantly. This allows the others to run around and fix windows, get weapons, and Pack weapons. You earn points for a while, but eventually the points from fixing the one window with the zombie will stop. Switch off with another person, and you can go get your weapons.
Luck is a major factor in your ability to survive to get 5 Pack'd weapons. By round 20 we had 3 people with Pack'd Rayguns, 1 person with the Pack'd Wunderwaffle, and everyone had Pack'd MG42's. Everyone managed to get at least 5 Pack'd weapons, and we also had enough points left to get through the teleporter 8 times. Using this method, it is also possible to survive until Round 20 without getting perks, knocking out another achievement.
The biggest thing to note:
You can Pack the same weapon!
I Pack'd an M1 Garand, Wunderwaffle DG-2, Browning M1919, and 2 MG42's.