Question: What’s the “formal” way to begin and end a letter or email to a Priest or a Bishop?
There is probably a bit more variety in how this has been done traditionally than this article suggests, but there is a useful article on the Orthodox Christian Information Center, with a lengthy excerpt from Fr. David Cownie's book "A Guide To Orthodox Life":
http://orthodoxinfo.com/praxis/clergy_etiquette.aspx
If you are being very formal, you would begin your letter by putting the date and the saint or feast of the day, usually justified to the right top of the beginning of the letter, but in an e-message, I think this is not necessary, though certainly not inappropriate either. As the article states, you would begin by asking for a blessing (for a bishop: "Master bless!", or for a priest ("Father bless!", or simply "Bless!"), and then begin with "Dear Vladika", or "Dear Father" or "Dear Fr. So and so". At the end of the letter, you could sign off with "Kissing your right hand, Joe Schmo," as the article suggests. You could also say, "Asking for your prayers, and kissing your right hand," or you could say "In Christ," etc.
As an aside, the book by Fr. David Cownie is very useful. I would recommend it, but only caution that he is writing from the perspective of a Greek Old Calendarist, and so he sometimes states things as being more cut and dried or uniform than they really are. However, the book has some of the most practical information I have ever ran across. For example, he talks about how to maintain an oil lamp, and how to get fasting food on an airline. It is definitely worth reading. You can get a hard copy by clicking here. You can read it online (in pdf format) by clicking here.