This achievement requires you to kill 5000 zombies in the menu mini-game. The bad news is the kills have to be in a single game (you cannot die or start over).
If you're going for all of the achievements, you might want to consider doing this along with the one for surviving the mini-game for 60 minutes. That's tedious and tiring, but will net you ~3000 kills and put you 60% along the way toward achieving this one. It's not absolutely necessary (you can get the other achievement without getting this one), but makes sense to kill two birds with one stone.
You'll find that the kill rate is pretty steadily ~50 per minute (or 250 per 5 minutes, 500 per 10 minutes, etc.). This achievement will therefore take approximately an hour and forty minutes of playing to get.
The basic strategy is to start by using the W, A, S, D keys to move to the center of the screen. After that the only key you'll press is the Q button to throw grenades (don't bother shooting the gun). You will spam the hell out of grenades: you have an unlimited supply, they are instant kills, they can sometimes take out multiple zombies if they're clustered close, and they can also take out zombies that are in close proximity to the grenade when it explodes so your aim doesn't have to be as accurate.
From the center of the screen, use the mouse to control the direction you're facing and just keep throwing grenades. After awhile you'll start to notice a circle/halo of blood surrounding your character. This basically defines the edge of the kill zone and you can use it to judge which zombies are getting closest to you. Preferentially target those inside the circle or closest to it. Undoubtedly a few zombies will get quite close to you over the hour; just patiently aim and take them out at close range.
After awhile your hand(s) will become tired, but there's little else you can do (switch up pressing fingers, quick flexes, etc.). The game does not get more difficult over time; there are some thicker and heavier waves of zombies, but they are just as frequent at the beginning as an hour in and the speed never changes.
There are a number of achievements you will get along the way: