Freelance writing was a career I kind of randomly fell into. I was working at a travel agency after my college graduation and was in charge of tour itineraries. The job was pretty much a yawn and I found myself with tons of downtime. To keep myself busy, I started to look into writing jobs I could do in my spare time. Unfortunately, my first freelance writing gig was working for a college essay website (I would not recommend this).
I began to apply for more article writing jobs and although most paid an embarrassingly low fee, it was a good way to build my freelance portfolio. No matter how talented you are, better paying freelance writing jobs are not likely to come your way if you've never been published before. Even if you feel your work is worth $0.25 a word, it's not going to happen for a newbie.
The first way you can build your portfolio is to submit work to a revenue share article submission site. Since most of these sites provide little to no editorial guidance, recheck your work to make sure you're highlighting your talent with the written word. Website examples include Suite 101, Triond and HubPages.
The next thing you can do is to sign up for an account through a freelance job website. Sites like eLance and oDesk allow you to fill out a profile and submit examples of your writing. You can then view and bid on writing jobs you're interested in. Once you land a few jobs, customer feedback will help you get more work and increase your bid amounts.
Lastly, consider a blog. Start a free blog through Wordpress or similar site. You can always submit the link to potential employers if they're looking to get a feel for your writing style.
As a new writer, you're not likely to make a ton of cash. However, taking a few low-paying jobs that give you a byline will help build your network and add more published articles you can use for future employers who wish to view samples of your work.