As @audionuma explained, if you are freelancing, you are effectively running a business as far as the law is concerned and you are expected to register accordingly. You also need to have a status allowing you to work in France, but as the spouse of an EU citizen you are covered in this respect.
Depending on the status you choose, you would also declare the additional income as part of your income tax return at the end of the year or pay income tax right away (prélévement libératoire) but that's not enough. The important difference are the mandatory tax-like contributions to the state insurance system (sécurité sociale, which covers health, unemployment, old-age pensions, work accidents and occupational illnesses, and family benefits) and corporate tax. As an employee, your employer takes care of all that for you but you are not allowed to simply circumvent the whole system because you're freelancing (or selling things abroad)!
“Auto-entrepreneur” is precisely a status to make all this (relatively) easy when your freelancing activity is limited (in theory, it's intended for people who are just starting or doing it besides a regular job, even though there is a bit of abuse going on). Unless your employment contract has a very restrictive non-compete clause, doing that beside a job should not be a problem at all (certainly not as far as the state is concerned). The main limitation is that your yearly turnover should be under a certain threshold (depends on the type of activity, for services it's something like €35k). If you exceed that threshold, you will have to switch to another status (with heavier bookkeeping requirements, proper VAT accounting and many choices to make regarding taxes and the like).
Concretely, you register on the official website, your registration will be transmitted to the relevant authorities and you will get a number of letters confirming that, a SIRET number, etc. After that, you can declare your freelancing income directly on the website and pay the relevant taxes and contributions automatically. You are out of the VAT system (i.e. you don't have to worry about it, you pay VAT like a private customer, you don't charge it to your clients) and you have limited accounting obligations (basically you need to issue invoices with a few key pieces of information – stuff like “TVA non applicable - article 293 B du CGI”; no proper double-entry bookkeeping).
It really is quite easy (and completely free, except perhaps sending out one or two letters, you only start to pay something once you have a contract/client) but the main issue in your case will be the fact that you expect most of your clients to be abroad. If they are inside the EU (and associated states), you are supposed to get some sort of special document attesting that you are covered by the French state insurance system (so your clients don't need to pay similar contributions in their country and/or can prove they are not trying to defraud their country's statutory insurance system by hiring a freelancer abroad) and I never managed that. After much back and forth to find out who could actually issue this document, it seems you first need a certain level of activity in France before you can get it and I lost an interesting contract in Switzerland that way. On the other hand, if your clients are completely outside the EU, maybe that's not relevant and you could just send them a regular invoice and be done with it.
A small note: Once you register, you will be listed on official lists of business owners, which are public. Because of that, you will receive a number of scam attempts, official-looking forms with a letter demanding a ~€100 payment to be added to some directory or other, which are neither mandatory nor really useful. Ignore them.