I was wondering if I can begin a more or less formal letter the following way:
Unfortunately, I didn't get any reply from you to my letter last week. (I) hope everything is fine with you..
Thanks in advance for your comments!
I would say for formal:
"Dear Sir/Madam (or their name if you know it)
I have not received a response to my letter of last week. [From here, the main body of the letter. In a formal letter, I certainly wouldn't ask after the health of the recipient.]"
"Dear [first name/forename]
I hope you're well. I wrote to you last week but I haven't heard back from you...."
Hope that helps.