Thanks for pointing that out. I assumed that DoD would not pay, but my omission might have given @ts1234 the wrong impression.
Also, @ts1234 might want to understand that priests are rarely (perhaps never), to my knowledge, permanently assigned as military chaplains. Each priest normally remains permanently assigned to his home diocese, and is a chaplain only temporarily. Again, I recommend contacting the Archdiocese for the Military Services for details.
I also recommend that @ts1234 consider exploring a religious order such as the Jesuits or Dominicans. These orders have their own priests, and being in a monastic environment has some similarities to being in the military. Also, such orders have a worldwide presence, which gives a broader perspective than that of a parish priest.