When writing a letter for someone, what is the best way to say so? Rather than simply use the words, "I am writing this letter for John Smith"
If writing a letter for someone else:
1) business letter - it is common enough for business letters to be written and/or signed by someone else. In this case, it can be acceptable to write the letter as if it were from the President of the company or another high level representative if you are authorized to do so. Put pp: or p.p. to indicate the letter was signed by someone else. (I'm sure we've all received plenty of sales letters in this format!) Here's thelawdictionary.com giving a little more detail about this practice.
2) letter from an individual - if writing a letter for someone else you should make clear right away that you are not that person and whether you have their permission to write:
Example A: "I am writing you on behalf of your customer, John Doe, at his request. John requests you..."
Example B: "I am writing you on behalf of your patient, John Doe, because he is unable to leave his apartment and refuses to accept help but his sickness seems to be worsening..."
Here's some general advice for a letter from an individual written for someone else.
I am writing this letter upon the request of my client Mr. J. Doe who felt unable to format his request into letter form. The letter is composed of information provided by Mr. J Doe but not a direct dictation.