These achievements will be earned throughtout the course of the game if you take out a majority of the enemies. Switch to another Primary/Secondary weapon once you have gotten it so that you can start raking up kills for another gun. Kills online also count towards these achievements (whether be it AI or player kills).
For the Assault Rifle, Handguns, and Sniper Rifles achievements, a good place to get them is the final training area of the first mission (Somalia). Ask your partner to advance ahead; he will then stop cause there is a proximity mine ahead. Throw a grenade beside him; he will run forward and be injured by the mine. Drag him around the corner and kill the guards that keep spawning. You have unlimited ammo and invulnerability. There will be 120 guards spawning 2 by 2 behind the fence so take them out at your leisure with one of the 3 guns. (This is because you are stuck with stock weapons for the entire first mission). Your AI partner will not die while you are doing this.
You can also get the Submachine gun achievement in the last area in training. Start the the first level Somalia with Salem, select your secondary weapon when the training tutorial prompts you to swap weapons, get a submachine weapon from Rios, and do the steps outlined in the paragraph above.