In my case it was pretty straightforward.
I wasn't a psychologist or psychiatrist, I was a Hospital Corpsman in the Navy (medical, kind of like a nurse), I went to what is called "C" school for training in psychiatry, and worked on a few different in-patient psych wards. The closest civilian equivalent would be somewhere between a psych tech and a psychiatric nurse.
As for how I got into that - I sort of fell into it. I did terribly in highschool, and grew up in a small, rural town. I didn't have a lot of options, so I joined the Navy.
I did well on the entrance tests, so was able to qualify for any job I wanted. The recruiter told me the best jobs were nuke power and hospital corpsman, I went with HM thinking it would be easier to get a civilian job afterwards.
I did so well on the exams, I was guaranteed a C school, after boot camp my options were pharmacy or psychiatry. Counting pills didn't sound interesting to me, so here we are.
This was all in the 90s, I'd been writing software and messing around with making games since the late 80s. Suddenly, in the mid and late 90s, writing code became highly in-demand. I bought a laptop (very rare at the time) and spent nights on the ward while the patients were sleeping teaching myself C, C++, and Visual Basic. I started moonlighting on my off time, and was able to get a full time job within a week of getting out of the service.
Been doing it ever since.