This isn't really a solution, simply a description of the akrid that you need to kill. I only did one play-through of the game on Hard difficulty and I don't know if the requirements differ depending on what difficulty you are playing on, but for me the requirements were;
1. 20 Sepia - Diaz's Cavern
2. 4 Dongo - Bishop's Wake
3. 6 Goonroe - The ravine area in Bailey's Crossing which can be reached via the caves opposite the machinery.
4. 10 Tarkaa - Abandoned Power Plant in Marshall's Gorge
5. 5 Wardeyee - Armory/Security Installation in Pickett's Folly
6. 1 Suwankaa - The first cave up the side of the mountain in Shackleton's Peak
(The locations listed alongside the akrid are the places I went to gain the kills.)
Every time I met one of the requirements I would return to Kovac at the next opportune moment and receive the weapon upgrade. I would then leave the lab and take the elevator/lift back up to the barracks and then head straight back down the elevator to the lab to speak to him again so I could receive the next set of orders this way I could potentially complete them on my next story related mission or as soon as possible.