Take as many freelance writing jobs as you can. Write content for Web sites, product reviews, and take any other type of writing job that you can use to hone your skills. The more you write, the better you will become at putting words together in such a way as to reach readers on a basic level.
Build a portfolio. Every job you do represents a client. Keep your portfolio updated. Include a list of your clients and details about the copywriting job you performed for each of them. Be sure to include actual clips of your writing -- as many as you can. Each clip you provide represents demonstrated copywriting ability. That may be enough to get you in on the ground floor of a corporation you want to write copy for.
Consider going to school. While excellent writing clips may get you copywriting work for smaller companies, higher-paying news jobs or ad copywriting jobs for larger corporations may still require a journalism degree or at least a degree in marketing, communications or liberal arts.
Take an internship at a newsroom or a copywriting firm if you can get one. Your portfolio and educational background, along with a good presentation at the interview, may help you get your foot in the door. At an interview, use the same techniques to sell yourself verbally as you do when you write effective ad copy.