Players are going to have to get five kills with each weapon through the CAMPAIGN. This is easily obtained through the first mission as the SMG is the first weapon given to the player and the rest are dropped by killing enemies. I managed to pick up the RPD (LMG), the Hauer 77 (Shotgun), and the AK-47 (AR) within the first minutes of the game. It can also be easily obtained through normal playthrough unless players are avoiding using each gun class.
Having gotten five kills with each weapon on MULTIPLAYER and ZOMBIES, I can confirm that the achievement is only obtainable through CAMPAIGN.