All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad, is His slave and Messenger.
May Allah reward you for your keenness on guarding your chastity and avoiding what is unlawful, and we ask Him to bless you with guidance, piety, chastity, and contentment.
You should know that marriage is a long journey, and therefore one should carefully choose his or her life partner. One of the most important criteria in choosing a wife is religiosity, acting upon the directive of the Prophet : “So select the religiously-committed one.” [Al-Bukhari and Muslim] It was also narrated that the Prophet said: “The worldly life is but a (quick passing) enjoyment, and the best enjoyment of this world is a righteous wife.” [Muslim] So do not settle for your previous knowledge of her from school, and rather ask about her the trustworthy people who know her.
If you come to know that she is righteous, then go ahead and propose to her Wali (guardian), and there is nothing wrong with the letter you wrote to him introducing yourself, explaining how you came to know his daughter, and expressing your wish to marry her. It would be good if you could find someone to be an intermediary between you and him. We advise you to seek the help of Allah, the Exalted, supplicate Him, and perform Istikhaarah (guidance-seeking) prayer, invoking Allah to choose for you what is good. For more benefit on Istikhaarah prayer, please refer to Fataawa 173969, 172573, and 81434.
Allah Knows best.