I am a French citizen and a permanent resident of America, and my American fiance and I are getting married. I would like to change my name legally in France as well as America. What is the process so that my name matches in both counties?
Your marriage certificate will allow you to change your legal name both in the US and in France.
After getting married, file Form I-90, Application to Replace Permanent Resident Card. Check “Yes” when asked if your name has legally changed since USCIS issued your most recent card and fill out the appropriate information.
After you receive your new LPR Card, update your Social Security Card and that will allow you to get a new Driver License and then change other documents.
As for France, I'm not familiar with the procedure as each country has specific rules. Generally speaking, you would go to the French consulate with your marriage certificate and ask them to register your marriage there and change your name to your married name.