This achievement is called Zodiac, for destroying all AC's of the Zodiac faction in the game. there are 9 of them to kill in the Order Missions, which can be done quite easily solo, or even in co-op, and if you are in the same squad, both players are credited. As for the other 4, there is one in missions 01 and 04, and two in mission 06.
Mission 01: you need to kill EVERYTHING, including the train, take less than 5,000 AP damage, AND do all of this in less than 5 minutes. A Zodiac AC will Break through the wall in the tunnel instead of the City-Police AC if you did it right. This is doable solo, but co-op helps double your firepower and kill things quicker, as well as spread out to get everything much easier.
Mission 04: I am unsure whether there is an AP damage prerequisite, my guess is ~5,000-10,000, but getting to the Executive Sector in like 3 minutes should spawn the Zodiac on this mission.I tried solo and got him first try, but couldn't do it in co-op with a friend. Maybe I sucked that time; no idea. But it's one of the easier ones to trigger, and he'll appear in place of the City-Police AC, but on top of a skyscraper, then jump down and try to rain hell on you.
Mission 06: Okay. This is where things are interesting. To trigger the two Zodiacs in this mission, you need to do 2 different tasks, which will take 2 play-throughs to complete. The first, you'll need to basically blitz through the level in like ~3-4 minutes, and I believe you cannot take more than 5,000 AP. It will replace the tetra-pod Migrant AC at the end.The second is somewhat different. You need to destroy ALL* enemies, and take less than 10,000 AP and beat the level (*ALL means the turret emplacements, sniper AC's and fast-moving AC's. You don't have to kill the Schit(defensive, shield enemies)AC's, but it does net you more money if you do, and keeps them for damaging you) Once this is done, a fairly similar variant to the first Zodiac in this level will spawn and then kill it just like the other. If this is your last Zodiac to kill, then you'll also be given the emblem Angie has on her chopper, which is also a great help for the emblem master achievement.
Now, as for recommendation to builds for tackling these suckers, i suggest anything with a fast generator, mainly a high-acceleration generator(like the UGN-70/Ho Vital), and a high-speed booster(such as the UBT-25 Gale), as well as parts with a high energy recovery rate. My AC had 5,964 energy recovery, and was on the Putuo heavy inverse legs for fast quick-boosting and jumping capabilities. As for the weapons...I used two USG chainguns and two Strekoza battle rifles, with the Fenugrec AD-10 reserve magazine for more ammo.
For some of these battles, like the encounter in mission 04, if you are killed by the Zodiac while doing the mission and think "Great...now I have to do all that all over again..." this one will actually save your time when you get your save progress, so you should have a second chance at killing him. As for the others, I think you get basically one try for them, but I cannot confirm this. I'm not very interested in going back through just to figure out "what-if's" when I've already done it. Besides, I need to focus on my squad a bit more now. If you have any questions, or something, leave a comment, I'll maybe get back to it somewhat fast, but I don't get on the computer much. But, if you really want to know something fast, it's best to message my gamertag, I'm online like everyday, so I can answer anything there much quicker.