This method involves using the Breakdown mode.
I highly suggest using Curubani if you have her unlocked. You can use another character if you want but Curubani seems ideal with the fact she has two swords from the start. In the lower left after you've racked up roughly 15% of kills (they've ruined the achievement tracker app so you won't be able to see the percentage) you should see a yellow sword icon pop up in the lower left above the radar, this means the weapon is about to break.
Pressing up on the d-pad will bring up your inventory and then you need to highlight the not equipped weapon (one currently equipped has a box with a little check mark) and press cn_X to equip the new sword and continue killing zombies. After about 30% progress from using both swords quit out and then restart the killing process. You can not switch weapons while in a car so make sure you're in a relatively safe area.
Avoid large groups of zombies. Five or more can be hard to handle. Getting stunned or grabbed wastes your time.
Avoid the special zombies, you can get progress off of them but it's not worth it. The military soldier zombies are fine for you to kill, they're just a nuisance.
Use your silenced gun to thin out large groups.
Use your twinkies if you need to continue fighting. Use your pills before you get downed the first time and if you get downed the third time just immediately quit out to restart and skip the death cutscene.
Swords, blades and daggers work for this challenge. Axes and other spikey weapons shouldn't.
Found out last year that the Buzzsaw works, it has like a spinning saw on the top of it. Just a heads up that some non sword-esque weapons may work.
This should take between half an hour to forty minutes.