Research what grants are available. There are books released annually on opportunities for writers, such as the "Writer's Market" series by Robert Brewer. The books provide information about foundations and contests dedicated to authors as well as listings of literary agents and different publications that can offer exposure. There are also websites that do the same, such as Michigan State University's guide to grants.
Contact your states art council. Each state has an arts council that provides specific information on grants. Often the acceptance of grant applications works in cycles. For example: Spring cycle, literary grants. This means you can only apply for a literary grant within that specific time period.
Complete the application process. Some grants and competitions will require you to submit your book while others will give you the money to write it. Read the applications for the grants very closely because if you don't follow all of the instructions perfectly, your application may be rejected immediately.
Write a grant proposal. This is a requirement for some applications and is also a way to find private funding. The grant you will write will explain what your book will be about, why it's important, what your purpose for writing it is, and what you're asking for to finish it. Grants van be incredibly challenging and many people hire grant writers to do it.