When you write to (work and application forms) will use the "私" essentially public documents, may also "わたし" in the novel, such as diary and will be used.　In Japan, children will learn the "hiragana" in the lower grades of elementary school from kindergarten. Young children may be written as "わたし" because it is difficult to write Kanji.

私 is correct. Writing its equivalent in Hiragana is quite vague. 私 can also be spoken as "watakushi" in very formal situations. When so happens, it is kept written as 私, or written as 私くし. In short; always write "I" as 私.

私 is the correct way to write "I" in Japaense. I私 is わたし written in kanji. わたし is written in hiragana. Both are pronounced "watashi" Use 私 only if you are female (or feminine, or really old, or extremely formal ^.^)

thank you for the explanation :)

I use both daily, like when I text to my friends, but I think you should write 私 in a formal situation, such as business letters, your resume/CV, an essay in class/for an exam.