Should I write in kanji?

If I were writing a letter (or something similar) in Japan, should I write it in kanji?

I don't quite understand your question - if you know Japanese and can write in a combination of Kanji, hiragana (one of the phonetic alphabet sets), and katakana (the other phonetic alphabet set), just as locals do, then by all means, yes.

If not, but if you can write Japanese sentences in "romaji" (romanized Japanese writing style), you can do that.

If you don't read/write Japanese, but if you were writing a letter in English to someone who understands English, you can write your name as it appears in your language and people's addresses in "romaji."