I am in the midst of writing a thank you letter to my French teacher. However, I am unsure of the correct way to start and end my letter. Is 'Chère Mme X' as a greeting and 'amicalement' as an ending appropriate if you are writing to your teacher?
Answer as it stands after a necessary correction by Stéphane Gimenez♦ (You shouldn't use endings such as "Je vous prie d'accepter/d'agréer…")
It all depends on the particular relationship that binds you to your teacher; if it's a formal relationship, the usual "Cher (Chère) Monsieur (Madame)," is recommended; you might add " le professeur" (Cher Monsieur le professeur,) but there is no necessity; not doing that is well too. You might also use a short formula such as a simple "Monsieur (Madame) (Mademoiselle),". "Amicalement" is excessive as a formula to end the letter. You need to say something that reflects more the distance and respect that the relationship implies. All sorts of possibilities are open; you might try to be inventive and end with something as for instance "Votre élève pour toujours reconnaissant," or "Votre élève reconnaissant,".
If your relationship is on the terms of a friendly enough exchange it is more difficult to say; in any case an ending such as "amicalement" will be proper. Of course if your teacher allows you to use his (her) christian name to address him (her), there is no problem with the use of that name in the opening of the letter, but that case will surely be among the rarest; therefore, the opening formula should be one of the usual ones, best fit for the case of those not too friendly relationships.
ADDITION: A vast choice of possibilities among the traditional forms used as proposed by dicocitations Le Monde.