Launch Dying Light and go to the EXTRAS menu. From there, select Xbox Store and see if "Be the Zombie" is available for download. It might take a minute or two before it shows up. When it does, initiate the download process and after installing the update, the "Be the Zombie" option should appear in the game's menu, as shown below.
Additionally, you can perform the steps presented in the following article: Black Screen - Clearing Cache.