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.