The same as you would to any Roman Catholic bishop. The salutation is either “Dear Bishop [last name]” or “Your Excellency.” The close usually is “Very truly yours,” although “Respectfully yours” is also appropriate. Since the letter is to a bishop, presumably about religious matters, you could also add “in Christ” to the end of the close. The same forms hold true for an English archbishop who is not also a cardinal.
Do not listen to anyone who may tell you that English Roman Catholic bishops and archbishops should be called “Your Grace.” That is an honorific given Anglican bishops and British dukes. While an English Roman Catholic bishop likely will not mind if you call him “Your Grace,” the proper form is indeed “Your Excellency.”
