Of all the weapon specific kill achievements this one will take you the longest. Personally I found Stage 2-5 the most convenient to farm for this as the balloon mines have low HP, don't dodge and there is an infinite amount of them if you keep the boss in Phase 1 indefinitely by not focusing him.
Setup: Get the main pilots active ability upgrade to spawn weapon crates ( whether you get the double crate or single is up to you. The double let's you reach the flame shot faster, but will inconvenience you more once you have reached it and just want a refill as you will have to dodge the extra box), then rank up your basic weapon to flame shot ( 2 upgrades needed). Fire by tapping the button or holding it briefly to minimize consumption. Once it's empty you will get downgraded to rapid shot. Spawn a new box and you will be at flame shot again after the pickup. Do not pick up any random boxes you may receive as drops, or you will upgrade to electric shot and have to waste it until it's empty again. Which plane you use doesn't matter but one with more HP will keep you in the fight longer. You don't need to finish the boss for the kills to count but the tracker will only update after you failed or completed the stage.