Start by doing research on your subject. Choose someone that flat-out fascinates you. You're going to be "living" with this person for a long time, so choose carefully.
Get the basics: your subject's place of birth, hometown, when they died, favorite colors-you get the idea. Then, trace the subject's family history, emphasizing any prominent family members and their particular contributions to the subject's life.
Outline major events in the subject's life: education, relationships and jobs, for example. Create a list of effects/impacts the subject had on society, or any historical significance the subject had.
Start writing. Resist the temptation to start with the subject's birth. Lead with an interesting moment in the subject's life to hook the reader.
Write a rough draft using your research as inspiration. Follow a chronological order of the person's life if possible. Include any noteworthy accomplishments, events, tragedies, successes, and more. If the person is still living, end with un uplifting conclusion about their path for the future. If they are dead, then conclude with one of their great accomplishments.
Read and revise your draft. Let your first draft sit for at least a month while your creative battery recharges. Then, look at it with fresh eyes and give the draft to a trusted reader. Listen to any comments the reader offers. Repeat this step as necessary.
Submit the biography to a publisher or agent for publication consideration.