To get this achievement, you will have to get a total of ten kills with any of the two Tank Destroyers together, the SPRUT-SD and the M1128, and ten kills with the two Mobile Artillery vehicles, the BM-23 and the M142. Thus, you can get 2 kills with the BM-23 and 8 with the M142 to be done with the Mobile Artillery vehicles.
If you have premium and have tested out the new maps you will have noticed that getting ten kills with an Artillery vehicle is going to be comparable to giving birth to a couple of moderately sized hedgehogs. This thing is basically a mortar with a range of 600 units. (Is battlefield measured in meters or yards, one might ask?) Only it's aimed like a grenade launcher and has a spread of a minor city with six missiles in a burst. You couldn't hit the ocean from a boat with this fucking thing, and it takes two shots to eliminate a tank. I did spend entire games shooting in the general direction of a flag without even getting a hitmarker...
Now since it's so hard to kill things with it and because of the fact that people always position it far away from action, it rarely gets taken out. Good thing right? No, this merely means that you have one or two chances to get it in an entire match before someone on your team hogs it for the rest of the game, forcing you to leave and find a new one or wait for the match to end and rush for it in the beginning of the next.
However, after hours and hours of frustration I started playing more casually and took a ride in the gunship. That thing is a raping machine from the seat with the heavy cannon and after fifteen kills with it you unlock an anti air missile for your puny little mortar. Suddenly you have a long range plane eater that nobody will even care about shooting at. Search for a conquest game with Bandar Desert as the next map in queue, wait for the current game to end and take the Artillery as soon as the match starts. Position yourself just after the little hill in front of your spawn area and go crazy on anything that can fly, happy hunting!