Doing work on you own time is an excellent way to show you actually WANT to do this rather than 'think it's neat'. Game programming really isn't like 'real world' programming. Getting into modding is something you can do today and there's really no magic 'do this and you're in' feat you can pull.
When I was trying to break in, I modded from the time I got home from work until about 2 in the morning. It was my second shift of work--5 days a week. Try to do something in a mod that is non-standard and proves you had to piece together how to do it on your own. You really have to put in thankless long hours (and have a understanding significant other), and if you stick with it, chances are you will make it eventually. Modding across several different games also shows an ability to pick up and run with code no matter what it looks like. Finally, scripting and doing design work with engines that require scripting is a big plus (for a variety of reasons). Some places will hire a junior programmer type to a scripting position.
Persistence is also a big key. If you really want to do it, keep programming, keep modding, and keep trying. Eventually your resume will land in the right place at the right time. It may take months or years (seriosuly), but if this is something you feel you want to do, eventually it will happen. Just be prepared to make a little less cash than you would 'on the outside'.