By far the hardest achievement in this game. I will break down the playthrough with the different sections. Unless stated otherwise, the recommend Power Ups are the Scatter Gun and the Flamethrower.
Section 1: Grass Zombies
There isn't really much to be said about this area, except that you must be moving. On your first attempt at this game you will most likely get overwhelmed, but after awhile you will quickly learn how to react to different amounts of enemies. In the long run, try to focus fire on the areas to allow you to get a clear path. Try not to walk to the edge because you can get cornered very quickly.
Section 2: Snakes and Slimes
The Flamethrower works great when you are facing Snakes and Slimes. The Snakes have a red face that is there weak point and with the flame thrower you can kill them instantly. You can walk through the white part of the snake as this is completely harmless. Slimes, when killed, form into several smaller slimes. If you use the flame thrower, you can kill the Slime without it even generating any more of the smaller Slimes due to how quickly they die. Eventually Zombies will start to come but don't focus too much on killing these as they only stick around for 20 seconds or so before disappearing. Not saying don't kill them, but try to watch out for the Snakes. When Zombies start to spawn it's best to try and stick with the Flamethrower or even the Machine Gun.
Section 3: Space Rocks
Your best bet is to not shoot any of the rocks unless you cannot avoid it. When you shoot one of the rocks it splits into several smaller rocks and speeds up. It's easier to avoid them and simply walk around for the two minutes it lasts. Try to pick up a Scatter Gun if you see one in preparation for the next section. If you have to shoot the rocks, then I recommend the Flamethrower and you can eliminate all of the particles before they travel too far.
Section 4: Exploding Zombies
This section occurs instantly after the previous with no warning with Exploding Zombies. They are exactly the same as the others but they explode when killed but is only a visual effect. Using the Flamethrower or Scatter Gun is best here as this is one of the harder sections due to the insane amount of zombies. If you're lucky you'll pick up lots of shields and Scatters Guns. Overall don't stay in one place and try not to die.
Section 5: Shape Attack
Shoot the shapes with what you have as there is no clear way to deal with these. They all take the same damage from every weapon so generally try to shoot them. There are also snakes here so be wary of those more-so than the shapes. If you can pick up a Flamethrower then you're pretty much sorted. Shields will make this section loads easier. Lasers, Shotguns and Rockets should be completely ignored at this section.
Section 6: Angry Faces
The faces aren't really a problem and die almost instantly to anything you fire at them. Scatter is still good here and you'll be able to take out multiple faces at once. Try not to get distracted by the giant cat face in the background that flashes occasionally. Also try to pick up a Scatter Gun or Flamethrower for the next section.
Section 7: Epic Outro
If you've made it this far, congrats, and prepare for the epic outro. With limited power ups and an infinite amount of enemies the only tips I can give here are Scatter and Flamethrower. Make sure to go out of your way to get shields also, as they can allow you to traverse through crowds of zombies.