You should start by sending a polite email explaining succinctly (researchers are always very busy: a short email will more likely be read) who you are and why you are interested in an internship in this researcher's lab (focus on what you can bring to the lab, and what you want to learn from them). The email shouldn't be longer than a few sentences. Attach your CV and a more detailed motivation letter: if the brief email manages to get the researcher interested, they will have more detailed material available right away.
Try to make sure your email will arrive in the early morning (often, people start their work day by scanning their inbox; emails they receive at other times can get buried quickly); which means take any time difference into account if you're not in the same time zone. A polite follow-up email a week later if you still have no answer at that time is OK.
Do not bring up that you will pay for your costs in the first email; this is completely irrelevant to the application (your application needs to focus on science). In France, interns must be paid if their internship lasts longer than 60 days (at least this was the case 10 years ago; I don't think this has changed), and large research centers often pay interns even for shorter internships. The internship stipend is insufficient to cover an apartment rental in most cities, but you can figure out practical details once your internship is secured. The researcher can point you to relevant resources to find affordable housing, but keep in mind that this is not their job and you should expect to figure this out mostly on your own. Some research centers or campuses might have guest houses. For example, Paris has the Cité Internationale Universitaire which provides affordable housing for visiting students. This is the kind of resources you want to look for, once you know you're going.