To figure out how much to charge, there are several ways:
1. Google and find people who are doing similar things to what you're doing, but in a different city. Call them and find out what they charge. For the kind of thing you're talking about, you could either bid on stuff on a project basis ("$100 to organize 3 closets") or a per-hour basis.
2. Ask your friends and family that you've done it for what they would consider a fair price.
3. Compare it to other broadly similar types of services in your city - luxury personal care services is the broad category, so consider things like having one's nails done, or massages, or hair salons, or home staging, or interior design. Those will give you a broad idea.
Once you get a ballpark range, pick a number and go with it. Once you start getting clients, raise or lower your price until you're working the 8-12 hours per week that you want to: Too many clients? Raise your price. Too few clients? Lower your price a little or, more likely, figure out why you're not getting clients. If your prices are too low people will assume you're not any good or there's some sort of trap.
Free services at first? Generally I would say don't do this, it has a tendency to get out of hand and you'll be doing too much free work and will struggle to switch to paying clients. Be straightforward and up front about what services you offer and what benefits the customers get from those services and what you charge, and people will respect you more and you'll make more money. Win win. Since you've already done it for family members, you can use them as references if anyone asks (ask the family members for permission to use them as references, of course) - no need to get any more references unless you find out you need them. For example, someone calls you and wants to hire you but asks for five references and you can only give them two. Even in that case, just give them 2, say you're just starting out. Even in that case, I would offer a 100% money back satisfaction guarantee. Most people will be reassured by that and will not ask for their money back.
Under the table: Legally you can't. It's not too expensive, but you should get a business license and business checking account before your first client and pay taxes on all of your income. In my state if you do a DBA (aka a sole proprietorship) with your name in the title "Suzie Q. Johnson Personal Organizing Services" that is free and no paperwork is required. Your local credit union or the bank where you already bank can probably set up a business checking for not much money if any. I'd suggest Quickbooks or something similar so you can do some basic bookkeeping.
You seem uncertain how to get customers. For businesses like yours, word-of-mouth is best. Tell all your friends what you're doing and ask if they know anyone who might be able to use your services. Do this repeatedly and it might take some time - you need to create awareness and then it takes a while for people to decide and actually make the phone call to you to get started. And tell all your friends and people you meet on a one-on-one basis when you see them. Tell the grocery clerk as you check out. Tell everyone you meet. Soon you'll get a customer or three; ask them for referrals, maybe offer a discount to them or their friend for the referral.
Finally, just start. People, including me, get all wrapped up in all sorts of peripheral things - business cards, business names, what color should the business flyer be, legal agreements, NDAs, etc. Just start organizing people's stuff, make them happy, and get paid. The rest will follow.
Good luck!!