This achievement is probably the easiest in the game and is pretty straight forward, like it says you just need to start up a game with each of the 4 weapons available to you.
From the main menu make a new game, choose your gun, to help you keep track you might find it easier to move from one side to the other, pick a level, once it's loaded up just press start and quit it, now, I made a save game for the first 3 weapons as a precautionary measure but according to other sources you don't need to do this, some people also say you should kill at least one person with each weapon, again it's not needed but feel free to. Anyway, when you're ready quit to the menu and and make a new game, choose the next gun and repeat the process.
I've linked in a video, unfortunately I couldn't find an Xbox recording, but it's the same process.
* Credit to KingSuperOne for the video