Take it from someone who sucks at zombies, this is easier than it sounds. In the menu of which zombies game type you want to select, press the right bumper and tab over to private session. Select Die Maschine Solo. I HIGHLY recommend the energy mine upgrade. I also took the XM4 with me but you can feel free to choose something else. I would however suggest something with a decent magazine size. Shotguns may be powerful but their range is limited and they need to be reloaded often. That could get you killed.
First things first, you have one self revive to start with so know that you can go down once and still succeed. Even better, you can kill some zombies while you are down and wait a little bit to revive yourself.
The first few rounds are pretty easy so familiarize yourself with the level and where the construction bench and ammo box are.
When you get scrap, use it between rounds to create stims. You'll need these if things get dicey.
As you approach waves 5, 6 and 7, zombies will become harder to kill and move faster so instead of killing a few as they spawn, simply run around until they have all spawned and grouped up. This is where the energy mine and any grenades or C4 that you might have picked up come in handy.
Don't worry about picking up double points or bonus points, you will get enough just from killing zombies.
During the middle waves, let the zombies spawn and form a large group. As you dodge them, stop, aim at head level and let it rip. One magazine a piece for each stop ought to give you plenty of time to start running again. REMEMBER THAT USING DEVICES AND RELOADING DO NOT ALLOW YOU TO SPRINT.
In the later waves of 11 and beyond, the zombies get even faster and more powerful. Instead of aiming when you stop, simply fire from the hip and don't worry about going through the whole clip. Also, keep an eye on your ammo as you may need to visit the box mid-round which is tricky. You should be visiting the ammo box between rounds whether you think you need it or not. Also, remember to keep your stims at max capacity as you approach wave 15.
If you go down and lose your self revive before wave 12, you may want to think about a restart. Remember that lethals are nice but other devices will replace your stims. The grenade and semtex are nice but the C4 and Tomahawk are better. C4 can wipe out a group and the Tomahawk is lethal in one shot to any zombie, good for taking out the really fast one's, and it lays on the ground waiting to be picked up on another pass.
In round 15, a Megaton zombie will appear. YOU DO NOT NEED TO KILL IT TO GET THROUGH THE ROUND. Be aware, that it has melee attack and a long range radioactive goo attack. The ranged attack will kill other zombies so it can be helpful from time to time. Do not be stingy with the stims. Your health can go pretty quickly this round. Once all the zombies are dead, the round will end and the achievement should pop.