1. Use polite verb forms.
Basically, Korean has two types of verb endings for polite speech style: 요(yo) and 다(da). When writing fan letters to celebrities, 요(yo) form is more appropriate than 다(da) form; 다(da) would sound too polite and formal for that kind of letter.
2. Use polite noun forms.
In Korean, some nouns have entirely different forms for plain speech and honorific speech. To be polite, use 저(jeo) / 제(je) instead of 나(na) / 내(ne). 저(jeo) is the equivalent to ‘I, me’ in polite speech. 나(na) is the casual form of 저(jeo). 제(je) is the equivalent to ‘my’ in casual speech. 내(ne) is the casual form of 제(je).
3. But don’t be too formal and too polite.
I’ve noticed that a lot of Korean learners tend to use the most polite noun forms when writing. Erring on the side of politeness is, of course, always good and safe, but in this fan letter, too-formal and too-polite nouns would sound a little awkward to native speakers (that is, the celebrities who will read your letter).