Update: You may notice either no progress pop up or even a delay unlocking this achievement.
Just get the twenty stars and then leave the game alone, it should unlock randomly a day or two later.
You can play any mode but I'll be playing Breakdown.
Now you can use any character but I'll specifically be using Curubani Kaur. (You'll have her unlocked if you purchased the game on the Xbox 360.)
Upon loading kill five zombies with either your gun or with melee
and you'll get a star. After getting both of those focus on only killing with melee to get two more FIGHTING stars. You'll probably unlock a REFLEX star while going for the three FIGHTING stars.
This process should take maybe roughly seven minutes depending on luck and amount of zombies that spawn around your starting area. A car is usually close by, so use that if you need to get to a area with zombies.
Quit out once you have the five stars and then start a new breakdown game and repeat the above. Upon your fourth or fifth new game you should have unlocked your 20th star.
This all together should take less than half an hour.
If you don't want to start a new game then feel free to check out the guides from the version of the Challenge that happened three years ago:
You can also try the guide from last year which will have you restarting the story mode, it may or may not be faster than my method: