If I start a Facebook page for my business, will people have access to my personal profile? Or should I create a new (fake) personal profile?
When you create a business page, it allows you to select yourself and other Facebook users as Administrators. While you can access your business page through your personal Facebook profile, others can’t—not even other Admins of your business page.
People who “Like” your business page won’t see that your personal profile is associated with the page and they won’t have access to your personal timeline. This helps keep your personal life separate from your business life, so you don’t need to worry about setting up a different personal profile.
If your business is your brand, simply use your own name rather than the name of a company for your business page, then you might consider making your personal profile harder to find, or completely hidden, so that Facebook users searching for you professionally don’t end up on your personal profile.