The lack of achievement tracking is frustrating for this one, but you can get a good idea of how many kills you have in dogfighting using this tip.
1) On the upper-left-hand corner of the main screen you'll see your pilot's avatar with the word "Social"
2 Open your own player profile and navigate to the dogfighting section. There are lots of stats here but not exactly what we want. Thankfully, we can derive what we want from what is here.
- Under Dogfight>Offense, record your "KD Ratio"
- Under Dogfight>Defense, record your "Total Deaths"
3) Simply multiply these two numbers to get close to your number of dogfighting kills.