Not too difficult this achievement, just requires a little bit of time. You level up weapons by getting zombie kills and headshots with them. Once you reach level 5 on a weapon, that's when you will unlock the final version of it. Simply purchase it with the ingame currency that you accumulate from missions.
To view your weapons and weapon levels, go to Customize > Weapon Customization and then you'll see a list of all the weapons in the game, with all the levels
On the latest patch you will now have to buy all previous levels of weapons in order to earn XP for the newest one. So, for example, you can only start only earning XP towards a level 3 gun, once you've purchased the level 2. Therefore you now have to buy all five of the level guns to get this achievement