This is a fairly organic achievement to unlock, and there's a good chance that you'll get it with simply putting in enough time playing Dogfight matches. The achievement requires you to shoot down four of each Starfighter, for a total of thirty-two kills. The Imperials have the TIE Interceptor, TIE Fighter, TIE Bomber, and the TIE Reaper. The New Republic has the A-Wing, X-Wing, Y-Wing, and U-Wing. You'll need four kills on each one.
If you're struggling to unlock the achievement, focus on destroying TIE Reapers and U-Wings. These are the least used starfighters in Dogfight as they're support craft, and the other starfighters are more appealing to most players. The U-Wings and Reapers are a little beefier on health and usually have decent missile jamming, so push more power to your laser cannons for that quicker kill. From personal experience, the U-Wing was my last starfighter to to shoot down for the achievement.
Holiday 2020 Update: The addition of the B-Wing and TIE Defender do not appear to have changed the achievement requirements. The B-Wing is in the bomber category alongside the Y-Wing, and the TIE Defender is in the fighter category alongside the TIE Fighter. You’re still looking for four kills of each category of star fighter class, which are Fighters, Bombers, Interceptors, and Support Craft.