You're going to have little to no time to work on your PPL. The only time would be on the weekends. You're going to need those weekends to destress and relax from your to NAVY flight training during the week. Probably not a good idea to get burnt out of flight training during Primary, also you'll probably be picking up bad habits from the civilian world, and switching sight lines and techniques from a T-34 to a Cessna when you have little to no skill in either is probably not a good idea either. You're going to have to get Civilian flight training ORM'd, which is probably not going to happen easily, especially if you are in the initial stages of Primary. The Opso or Stucon Officer is going to give you a crazy look. Not to mention the fact that going back to a Cessna or a Warrior is going to be a kick in the balls after flying a T-34 (think piss poor performance, no A/C). While it is a novel idea, you'll be fronting serious cash. Keep in mind if you want to take your friends up back home, you'll need a 1-3 hour checkout at the min, as per FBO insurance requirements.
This is on top of whats already mentioned in the prior posts; money, time, and orm wise, its probably not going to happen. Several of my friends in advanced, along with me, have our CFI certificates, and none of us has even bothered to fly on the civilian side since they joined the Navy. Just something to think about.