Mass Effect 3

As Alderon said, you can first modify your weapon in the second mission (I don't wanna spoil anything even if it is minor, but it's the the mission after the Prologue). While this is a bit difficult to miss, I wanted to go into a bit more detail and there's another mod to be found before getting to the one he mentions.

1. There is an SMG Scope mod (accuracy) when you enter the dark, depressurized cafeteria-like room in the base. As you're walking to your goal and heading down the small steps, it's on a counter top on Shepard's right side.

2. The second mod is and SMG materials which reduces the weight of the SMG by 50%. It is found in the same room as the Weapon Bench near a dead body and some respawning ammo (if you run out of ammo in the battle just ahead, you can come back here and grab the thermal clips laying on the floor).

3. The third is after you fight some enemies and make your way to the Decontamination Room. You'll find a Sniper Rifle Extended Barrel I mod that increases SR damage by 15%. You can run back to the Weapon Bench to install this mod if you wish. It's not that far away, though weight reduction or rate of fire mods would've been more helpful.

There are two more mods on this mission, but you can't access a Work Bench after you collect them.

++Modification Process++

1. Walk up to the Weapon Bench and press A to go to the weapon loadout screen.

2. Select the character you want to modify the weapon for.

3. Move the thumbstick to the weapon you wish to modify.

4. Press X to go into the modification screen.

5. Move the thumbstick right (or left) to select the mod you wish to install.

6. Press A to confirm the mod.

It seems a bit redundant to go through the actual process of modifying the weapon, but there is so much information on the screen at once that it can be confusing (especially when it comes to selecting the mods) so I thought I'd clear that up.

++Non-Solution Related Advice++

One last thing that has nothing to do with the solution and you don't have to read this.

I'd recommend removing at least one of your weapons at the Weapon Bench. Reason being is that weight of gear carries a burden in ME3: it slows down the recharge (or cooldown) of your powers. As an example, my soldier Shepard's cooldown for Adrenaline Rush was a lot slower than when I was running around in the Prologue. I removed the shotgun as it was -150% weight reduction and my recharge rate of Adrenaline Rush is much faster. The shotgun is powerful (at close range), but it's heavy and I prefer pinpoint accuracy so I kept the Sniper Rifle (very heavy) and Pistol (not so heavy). I'm guessing all classes have this, but, as a soldier, weight reduction can be bought via stat points or available to anyone through weapons mods.

That's it. Good luck on your journey through ME3 and enjoy the game!

EDIT: Sorry about the double post. I don't know what I did there.

EDIT #2: I said earlier that weight reduction mods can be found in ME3, but I noticed that it is misleading. The only weight reduction mods are available for the SMG (level 5 = 90% weight reduction of the SMG). The weapons I carried throughout the entire game on Insanity were the Viper Sniper Rifle V, the Vindicator Assault Rifle V and an SMG once Shepard found one in the game. My total weight was still at around 120% and it gave a nice balance between fire power and acceptable cooldown for my powers.

During the second story mission (unmissable), you will come across a Weapon Bench, and there will be some crates on the floor near it. Behind these crates there is an SMG upgrade. Pick up this upgrade and then apply it at the Weapon Bench for the achievement.

If you miss this one, the next Bench will be on the Normandy in the Shuttle Bay, and you can purchase some inexpensive Mods at the Normandy store also located in the Bay.

During the second mission (MARS: PROTHEAN ARCHIVES) you'll come across the M4 Shuriken SMG in the shuttle control room. The two mods you can get for it are the SMG Scope at the kitchen and the SMG Light Materials which is in the small room with the bench. Then simply use the bench and apply the mods and the achievement is yours. Video shows where all the items are.