Research labs are very international so as long as you speak English you can do a PhD pretty much anywhere. It's also pretty easy to get a visa for a PhD or a postdoc. As for the process, basically you need to hold a master degree and find a professor/researcher with enough money to hire you. Labs often advertise PhD positions on their websites.

Finding a grant/supervisor is quite easy for a strong candidate. The critical part is too choose wisely the topic. Research tends to be very specialized and you may end up with a PhD that gives you few marketable skills.

I'm a German citizen, and I learned French in school, my level of proficiency was very low when I arrived though. Getting around using English was quite easy, and the application process was straightforward even with a German Diplom (M.Sc.). For an US degree you'll just have to make a translation and possibly have it's equivalence assessed, that shouldn't be too hard though. The UPMC (Paris VI) has a quite good PhD program for example, and is open to international students.