This achievement can be done in single or multiplayer and can be done over many different missions (i.e., not in one mission). Make sure you buy the original handgun you are given. Picking up a gun does not count. You only need to buy the attachment for one gun, not for every gun.
What you need to get:
SUB MACHINE GUNS:
Barrett .50 Cal
Red Dot Sight,
M203 Grenade Launcher (e.g. M4A1),
M320 Grenade Launcher (e.g. ACR),
GP25 Grenade Launcher (e.g. AK47),