How to (legally) work as a freelancer in France when you are a french citizen ?
The easiest path is to go to the auto-entrepreneur status. Minimal paperwork. There's a limit on the maximum sales revenue you can make while on this status. You can nevertheless change status if needed (like if fortunately your sales go above the maximum for auto-entrepreneur).
Can I just declare my income ?
No, you can't. Working in France implies some social taxes (retirement, health, ...) that must be payed before the income taxes stage. It means you need some sort of legal status to actually work in France as a freelancer. You will then declare your income for income taxes purposes (with most social taxes taken out).
What if I am an artist ?
In France (as in most of EU I think), there's a special status for 'artists'. What's an artist ? It is someone who is mainly producing art stuff ;-).
Practically, if you are a graphic designer, making a logo for a (client) company, you can either sell them your work and give them all rights to publish, modify, do whatever they want ... (then, you are not an artist) or just give them the right to use this work for a dedicated purpose, asking for more money if they use it in another purpose or context (You are then an artist). There are really different rules in Europe vs USA in this subject, so if that is your case, you should carefully check contract terms.
But in France, it means you will have a different status (artist vs freelancer). As this is related to how much of your income is from 'art' or not, and as current clients are from USA, I would suggest to go into the non-artist path and adapt after a year.