DOOM

To have the chance to master a weapon you must first unlock said weapon and then unlock all the mods and then apply all the upgrades to each mod. The only exception being the Super Shotgun which doesn't have mods but still needs upgrading. You unlock a mod by finding a field drone. To get a guide on where to find all weapons and field drones check out the guide for this achievement. You will find what you are looking for there.


Now once you have unlocked a mod and upgraded it fully using your upgrade points you then need to complete a challenge to master the mod upgrade. Each weapon apart from the Pistol and Super Shotgun have 2 challenges to master the weapons mentioned have just 1.

Ideally you will want towards as many of these during your first play-through of the campaign. Personally I only had 4 challenges left to do after I'd finished the story. Once you finish and you need to grind the remaining challenges the majority can be done at the start of Mission 9 - Lazarus Labs. From the beginning point of the mission you just have to walk through 3 doors jump down some sort of elevator shaft where you will be met with around 10 weak enemies where you can boost your challenges for. Occasionally you might get a checkpoint you don't want, if this happens just quit out and start the mission again, this isn't really an issue as the method is right at the start of the mission. Here is a list of weapons/mods and challenges and the best place to boost them.

Shotgun
Charged Burst - Kill 5 Mancubi with a charged burst. These are the really big round fat guys. Chances are this will be one of the first mods you unlock so can easily obtain this throughout your first campaign run as these guys are everywhere. If you still need it any of the later levels have lots of these guys. Try mission 11 - The necropolis.

Explosive shot - 20 direct hits on imps. These are the small guys which usually hang back and throw fireballs at you. There are so many of these in the game its probably impossible not to have this unless you literally didn't use this. Mission 4 - Argent Facility if you need to. There are loads of these guys in the first 2 battles. Just beat them and restart if you need more.

Heavy Assault Rifle
Micro Missiles - Stick 4 individual demos in 5 seconds. Easily done on the mission 9 method stated above. Just spray your missiles between different enemies shouldn't be a problem.

Tactical Scope - 50 head shots while in tactical scope. Mission 9, one shot head shot on most of the weak guys.

Pistol
Charged Shot - 15 head shots with the charged shot. Mission 9, easily boosted.

Plasma Rifle
Heat blast - Multi-kill with the heat blast 20 times. Must shoot to charge up the heat blast the press to release the heat wave. Mission 9, shoot a few bullets off jump down, blast away then shoot away from the enemies to charge it up again.

Stun Bomb - Kill 30 stunned demons. will stun them you then have to kill them while they're stunned. Mission 9 best place for this.

Chaingun
Gatling Rotator - Kill 5 demons in 5 seconds. Pretty self-explanatory, mission 9 again just maybe wait a few seconds for them to group together more for a better chance.

Mobile turret - Kill 4 or more demons during a single deploy 5 times. Once again mission 9 is your best bet just kill 4 of them before you overheat.

Gauss Cannon
Precision Shot - Kill 5 Hell Knights with a precision bolt head shot. Hell knights are the big guys that will come charging at you to melee you. They look like beserkers from Gears of War if that helps. I recommend Mission 4 - Argent Facility. From the start of the mission the first major battle you have will feature 2 or maybe 3 of these guys. Try and get them into glory state then aim up the shot to the head. Then either carry on with this mission as there are loads more or restart to get the same guys from the beginning.

Siege Mode - Kill 3 or more demons with and siege mode shot 10 times. Easily done at the start of mission 9.

Rocket Launcher
Remote detonation - Kill 3 or more with a single detonation 15 times. This works by detonating the rocket in mid-air after its shot. Mission 9 for this one.

Lock on burst - Kill 5 summoners with a lock on shot. Summoners are the floating angel like enemies with a red light following their trail and they can teleport during battle. Mission 4 - Argent Field play the objectives where you are destroying the 3 argent consoles or whatever they are. One of these will be inside a big building which is lit up blue on the inside. The initial battle won't have one of these guys but once you've done the first battle and destroyed the objective, you will go back into the room to find one spawn right in front of you. Blast it away with your lock-on shot and reload the checkpoint. It will load to the point where it will spawn right in front of you.

Super Shotgun
Kill multiple demons with one shot 30 times. Just line up the shot with 2 guys close together. Easily done on mission 9.

Now this achievement is a bit of a grind, but not impossible. In case you’re new: after you purchase every upgrade for a weapon mod, you’ll unlock a Mastery challenge, which will ask you to complete a challenge that’ll lead into a major bonus for that mod. This can be increased damage, faster firing rate, etc.

For a quite of few of these, you’ll be able to grind them easily on weaker one-hit/two-hit enemies like Imps and the common Possessed for multi-kills.

Here's a full list of every weapon and the mods you'll need to complete the challenges for:

Combat Shotgun:
- Charged Burst: Kill 5 Mancubi (the big fat demons) with the Charged Burst to unlock the Power Shot bonus. If you aim for the head and unload three times, it'll usually kill him easily.

- Explosive Shot: Score 20 direct hits on an Imp using the Explosive Shot to unlock the Cluster Strike bonus. Simply aim at the head and fire; it’s usually a one-shot deal. Considering how little health they have, this shouldn't be much of an issue.

Heavy Assault Rifle (HAR)
- Micro Missiles: Stick 4 individual demons in five seconds with the missiles for the Bottomless Missiles bonus. There is an excellent spot for this; right after you acquire the BFG, you'll come across a massive horde of Possessed. If you've got them, switch to your Micro-Missiles and let 'em have it. You're bound to hit plenty of them to pop the challenge.

-Tactical Scope: Get 50 headshot kills while using the Tactical Scope to earn the Devastator Rounds bonus. This is self-explanatory, but you can grind this on every other type of enemy, considering that the HAR already has a massive damage bonus from the Skull Cracker upgrade.

Pistol
- Charged Energy Shot: Get 15 headshots with the charged shot to earn the Increased Power bonus. Like the bonus before, you can grind this on weaker enemies like the Possessed or Imps.

Plasma Rifle
- Heat Blast: Score a multi-kill with a single Heat Blast 20 different times to get the Heated Core bonus. For this, ditto on weaker enemies.

- Stun Bomb: Kill 30 stunned demons with the Stun Bomb to earn the Chain Stun bonus. Simply release the stun wave and kill the stunned enemy to get the bonus. You can grind the hell out of this on the Pinky demons, who not only stagger easily, but have a massive weak point in the form of their tails. Just make sure that the enemies are still stunned when you kill them.

Chaingun:
- Gatling Rotator: Kill 5 demons in 5 seconds while using the rotator.

- Mobile Turret: Kill 4 or more demons during a single Mobile Turret deploy 5 times. With the chaingun's charging crosshair, all you need to do with this is get 5 kills in Turret mode before the crosshair fully turns green. You can still keep firing, just don't overheat and you can get more progress towards the challenge.

Gauss Cannon
- Precision Bolt: Kill 5 Hell Knights with a Precision Bolt headshot to unlock the Volatile Discharge bonus. Self-explanatory, but weaken them until they've got a Glory highlight, then finish 'em with the Gauss.

- Siege Mode: Kill 3 or more demons with a Siege Mode beam 10 times. The beam tunnels through groups of enemies, which hopefully gives you a better idea of how it generally works. You should keep this mod after you upgrade it; it works beautifully on Cacodemons and Mancubi.

Rocket Launcher:
- Remote Detonation: Kill 3 or more demons with a single detonation 15 times. You can still grind this in the post-BFG room with the Possessed groups.

- Lock-On Burst: Kill 5 Summoners using the Lock-On Burst.

Super Shotgun:
- N/A: The Super Shotgun doesn't have any mods avaialble from field drones, but the challenge for this is to get 30 multikills from a single trigger-pull of the shotgun.
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Please do comment or state any issues before downvoting, and Happy Hunting!

Edit: Kudos to both Mr McMuffen and Dwaggienite for the post-BFG room.

The solution above tells you everything that you have to do and its helpful, but the location i chose to do the challenges was in the mission "Necropolis". In the beginning of the level, you have to kill a couple of enemies and then go interact with the skull and when you do it opens the door to the room where you get the yellow skull, its impossible to its because its the way to go for the story.

When you get to the room with about 7 or 8 basic demons(The Possessed) its perfect for most of the weapon challenges. For example, the siege mode for the gauss cannon where you have to kill three or more enemies with one shot, or the remote detonation with the rocket launcher where you have to kill three or more enemies with a single detonation.

Just make sure you don't accidentally explode the barrels to kill them and leave one possessed alive so that you can keep reloading the previous checkpoint. I usually go to the right side where there are little to no explosive barrels and i kill the 4 possessed on that side with whatever weapon. Just keep grinding it until you're done.