If you are moving to France permanently and bringing your car you will need to firstly notify the DVLA, you can do this by filling in section C of you V5 Registration Document and returning by post to either your local DVA office or the main DVA office. You will then recieve a Certificate of Permanent Export or V561 if you neglected to do this you will need to contact the DVLA and explain the situation to them they will normally issue you with the export certificate via post.
For further information visit the DVLA web site:http://www.dvla.gov.uk/vehicles/
Next you will need to visit your local tax office or Centre des Impots taking all you vehicle documents and proof of identity with you. Ask for a Certificate of Acquisition, this is just to prove that the vehicle belongs to you and that there are no taxes due on it.
You will now need to get your vehicle a Controle Technique (MOT) if your car is more than 4 years old.
Nearly every town has a controle technique center just look in your yellow pages for "controle technique de vehicules" or onlinehttp://www.pagesjaunes.fr/
Next step you may need a Certificate of Conformity - this just states that the vehicle passes EU safety standards, to receive one contact you cars manufacturer, some manufactures charge a fee for the certificate.
By this time your probably thinking of giving up and buying a French Car - but your nearly done, the good news is that there is no road tax in France and a Controle Technique is only required every 2 years.
Next you will need to visit your main Prefecture or Sous-Prefecture (normally in your main town), you will need to take all your vehicle registration, controle technique, certificate of conformity and import documents (if you didn't apply for these before leaving the UK you will need to contact the DVLA and ask for them). You will also need to provide proof of your residence & identity i.e. passport, house deeds or lease agreement, electricity or gas bill, insurance certificate etc.
The application can take several weeks and eventually you will be asked to return to the Prefecture to collect your new registration documents or Carte Grise (it's probably best to take all your documents with you - just in case). Your vehicle will be issued with a new registration number and you will have to have new number plates fitted, most garages can arrange for the plates.
Useful Information & Further Reading
Tel: 0044 (0)870 240 0010
Service Public - the French government website
The French Ministry of Transport
The French Customs & Excise website (English Version)