PharmaT, I very much doubt that you have offended anyone. I think you should apply the same strategy in a letter to an unknown relative as you would in a letter accompanying a job application. As they say, you only get one chance to make a good first impression.
You do hear of people who write wacky job applications who get the job just because it was right for the moment, but I don't think you can be too polite or too well presented, but a lot depends on who the relative turns out to be. If I got one full of spelling mistakes and smiley faces, I'd guess that they wouldn't be someone I'd have much in common with, but everyone is different, or at least 1% of their genes are!
It is challenging to write a letter to someone, not knowing whether they are a distant great aunt, or someone the age of grown up grandchildren, although the DNA data can give a clue.
Martin