Qualifications
Grant writers usually need a four-year degree (B.A. or B.S.) in English, communications, marketing, or any field that gives experience in writing. Writing ability is key; if you have a degree in Biology but can make that proposal sing, you'll be hired. Certain groups, like scientific research institutions, may actually prefer you have a background in their field.
You can go on for a master's degree or a doctorate, but it usually isn't required unless you plan on climbing the grant writing rungs all the way to the top. If you decide on that route, an MBA or a master's degree in nonprofit management may be your best option. Concordia University in Chicago even offers a Masters of Arts in Grant Writing, Management and Evaluation. How perfect is that?
If you want to pad your résumé even more, you can get certified in grant writing. It's an extra cost: coursework and certification exam can total upwards of $500 (source). Many professional grant writers seem skeptical of shelling out for credentialing, and in the end, winning a grant should be proof enough that you know your stuff.
Grant writing experience is almost always the number one requirement of prospective clients and employers. Welcome to the chicken and the egg of the working world: how do you get experience if no one will hire you until you have experience?
Luckily, there are ways to work around that. You could do an internship in grant writing (and maybe even get college credit for it). Worst-case scenario, you could volunteer to write grant proposals for a nonprofit you're passionate about. They always say you should do for a living what you would do for free. (Ah, the ubiquitous "They." Just who do "they" think "they" are, anyway?)