Becoming a full time portrait artist is just a matter of declaring yourself to be one and spending all your non-painting time telling everyone that you are one. Just joking, but only partly. Since there is no license or certification to "become a real artist", whenever you say you are one, you are.
As to the full-time bit, I assume you mean, how do you make more money in order to make a living at it. That's where the marketing comes in.
There are many tips on this forum about how to promote yourself as a portrait artist, and tips on the
www.asopa.com (American Society of Portrait Artists) website also. In addition, there are many good books about marketing your work as an artist, some of which are listed in the book section of this site.
Basics include getting business cards made up and giving them to everyone you know, telling everyone that you're available for commissions in case they know of someone who might be interested, getting a website, sending press releases to local papers, doing demonstrations at events in upscale areas (the opening of a new library, school art nights, etc.) donating to charity auctions, carrying around a pocket sized photo album with excellent photos of your best work and showing it to everyone, etc....
Before long many people will know you're really in business, and if your work is up to par, commissions will come. Being a portrait artist is about being an entrepreneur and like anyone who runs a small business, requires many skills and effort beyond just doing the painting.
Good luck!