Read a few other biographies of other artists. Reading other biographies is a good way to develop ideas for your own. The one element artists' biographies share in common is that they are relatively brief. The goal is to condense and summarize your artistic background in a relatively short space, usually one or two paragraphs.
Make a list of the essential things you want to include in your biography. It should include such things as your artistic influences, your educational background and training, a short synopsis of your work and a short summary of your philosophy of art. Your artist biography should give the viewer an insight into your background as an artist as well as providing a few clues about how to interpret your art. Let the viewer know what your motivation is and what you want to accomplish as an artist.
Write the biography using a list as an outline. Study your list of influences, artistic ideas and experience and condense the essential information and details into one or two paragraphs. Write two or three versions of the biography. Allow it to sit for a day or two. This allows you to evaluate the biography with a fresh attitude.
Re-read the different versions of the biography. Determine which one you prefer or write a new version that synthesizes different elements from the older versions.
Proofread the final copy of the biography. Ask one or two friends who are familiar with your work to proofread it as well. Ask you friends if the biography accurately describes you and your work and use their suggestions to fine tune it.