Similar to the achievement:
This one can be got by playing through the Prologue after completing the game and using the Relic which gives you early access to the Crossbow and Assault Rifle.
Use the Bolt Action Rifle in the first section of the game, then the Assault Rifle exclusively in the section where you must collect Gas Tanks and run for the car. Note that the Crossbow does
not count as a firearm. Using it will void the achievement.
Make sure to kill at least one walker with either gun. Don't let yourself get grabbed as using the knife for the execution will void the achievement. If you are grabbed, just let yourself die and restart from the checkpoint.