You must simply use one of the shutguns in the game and kill 250 enemies using any of the shotguns available.
The following weapons are shotguns: M3 Super 90, FSPS-12, and the good ol' DAO-X.
All kills count. You can get this playing in single player (co-op/AI partner) or in multiplayer mode. This does not have to be 250 kills with the same gun, any combination of shotgun kills will count.
If you make sure you swap out weapons types when you get achievements you will get this during one of the story play throughs.