The achievement is not as hard as it is made out to be. To finish most of the missions you only need to make sure that you get mainly headshots, and multi-kills if possible. This achievement can be obtained on easy, which I would recommend. Below is a video showing where to find all the collectibles for the game. I won't state this on any of the missions, but please use your right bumper to mark as many enemies as you can. Knowing where they are will save you a lot of time and heartache. If you want more points, remember that having your health in the red while killing someone gives you an adrenaline kill bonus, which adds 50 points to your score. Headshots net you 125, as well as Melee kills, so try to go for those above all else.
Also keep in mind that you will need to unlock all 3 endings in order for the achievement to pop, since each ending has its own assignment.
Baku: Requires 6,000 points for the assignment. Easiest way to do this is to go for headshots only. Having a rifle handy will make this a lot easier. I recommend the .338 or the Scout. This level is mainly infantry so headshots are the way to go. You can get a little over 7,000 by headshots.
Shanghai: Requires 10,000 points for the assignment. Make sure to shoot the first guy when the elevator opens on the wrong floor to net a quick 100 points. Headshot as many enemies as you can. Once they begin to come through the windows, you can either continue to shoot them in the head, or lay claymores beforehand if they overwhelm you. Once you're on the roof you can get a stab on the guy outside of the left door, and then just proceed to get headshots. after the second helicopter takes off do the same. When you head outside the building make sure to kill everyone, and wait for the helicopter to drop off more troops. Rinse and repeat until you get the Tanks. 2 sticks of C4 anywhere on them will kill them, so keep that in mind.
South China Sea: Requires 10,000 points for the assignment. Same deal as before, headshots above all else. Once you get into the boat, begin gunning the boats down around you. After the first wave, you can use the secondary fire on the boat to use lock on missiles to make it easier to get kill streaks going. Most of your points will come from the boat, so try to stack killstreaks if possible. The rest of it should be simple enough, headshot the infantry and use lock on missiles to take out the transport helis after they drop off people to maximize points.
Singapore: Requires 19,000 score. Easiest method I've found is to use the Scout Sniper Rifle to score headshots, and only use the tank to destroy other enemy armor. Hoof the rest of the way scoring headshots and you should finish the mission with a little over 19,000, or as I finished it, 20,125. At the end when you are positioned as the gunner for the Jeep try your best to kill everyone coming out of the vehicle to net 20,000 and the Storm Bringer Achievement.
Kunlun Mountains: Requires 11,000 score. Make sure to kill everyone with melee kills at the beginning, and try not to get spotted. After you flip the switches, this part can get tricky due to the lack of spotting. If you can pull it off, melee kill as many people as possible, then get headshots if possible. Alternate between melee kills and headshots to your liking. Once you meet up outside of the main prison compound, swap back to a rifle and mark the people on the bridge and kill them with headshots. once you're crossing the bridge, beneath it will be the .338 rifle, so switch to that if its to your liking. C4 the vehicle on the bridge, and then proceed to kill off all the enemies on your way to the gondola. You should have enough points by the time the last squad gets off the gondola, you could toss a grenade and kill some people for a possible multi kill if you still need points.
Tashgar: Requires 14,000 score. C4 the tank before engaging with any enemies, and make sure to have a rifle handy. This should make the engagement a bit easier. After you get to the second location, head right and melee kill the enemies by the three story building on the right, then take out the tank for breathing room. On your way to the dam you can melee kill the first two enemies then snipe from the ridge downwards. Just be careful of the planted MG's. Use your MG32 to take out the chopper when it comes to attack you, then proceed to kill the rest of the enemies for the level with headshots.
Suez: Requires 5,000 score. Headshot all the enemies, then get the MG32 to kill the helicopter that passes by after it drops off the enemies. Head to the right and kill the enemies on the lower left, then head up towards the right and kill the enemies as well. eventually the enemies will try to retreat, so kill the stragglers with the MG32 to make sure you get as many points as possible. finish off the second wave of enemies and kill the helicopter with the MG32 or whatever you wish to kill it with. On your way to the ending, allow the gunner of the boat to kill incoming enemies to net extra points.
Hope this helps anyone out who might be having issues with this achievement.
Edit: In case anyone is having missions which are showing a missing Dog Tag or Weapon, and you are sure you have collected them, the achievement should pop regardless of this. I had two missions (Suez and Kunlun Mountains) which said I was missing a dog tag on each, and once I finished the Singapore assignment the cheevo still popped.
Update: According to O R iii O N (SaidaiKanpeki), if anyone is having issues unlocking the achievement, get all 3 endings and complete all assignments then replay the mission the collectibles glitched on and you should hopefully pop your cheevo. Thanks to O R iii O N (SaidaiKanpeki) for the tip.
Also to be noted, Ryot Control has stated that he reloaded kunlun mountains and it unlocked for him. Hope any of this helps.
There are also a few comments in the comments section that might help out for some of you, depending on your situation. It seems like everyones game can glitch differently for this achievement.
Update: Some of my friends have been reporting that you can get random achievements in Test Range from just loading into test range. One of my friends recently got the Full Arsenal achievement this way. He told me that its a random occurrence but he earned 4 achievements through the test range, which were the ones he was missing. This seems to work for campaign related achievements only. If this works for anyone please comment below to confirm this. So far from what I've seen I can confirm that the challenges for the missions will unlock this way, as well as the collectibles, but I haven't seen anything for the campaign difficulties, or at least not from my friends.