First of all, it is exceedingly dangerous to make a diagnosis via media reports of a sensationalized event.
That said, the development of paranoid delusions can be very much a part of complex PTSD. The statements attributed to the killer would be consistent with that particular variation of a paranoid delusion. While tyme is correct that the killer was in the prime risk age range, there is generally a long, slow decent into schizophrenia starting several years earlier. While this is not always the case, an abrupt breakdown in reality testing is more Hollywood than reality. In addition, there is rarely a proximate cause such as combat. Extreme stress can cause a brief reactive psychosis, but this is usually a circumscribed event with full recovery coming quickly. In the end, there is not enough known about the killer's pre-morbid condition to warrant making the call one way or another.
Again, with caveats because the media reports are the source, it sounds as if the killer had on at least one and possibly two recent occasions been involuntarily hospitalized. If this is the case, then he would have been a "prohibited person" with respect to buying, possessing, or USING firearms. For me, it begs the question of why the family didn't inform Chief Kyle and his friend of this status.