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, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
We ask Allah, The Almighty, to guide this girl to the straight path and to embracing Islam, and to facilitate your marriage to her. It was narrated on the authority of Ibn ‘Abbaas that the Prophet, sallallaahu ‘alayhi wa sallam, said, "There is nothing like marriage for two people who love each other." [Ibn Maajah]
Your keenness on guiding this girl to Islam is a good thing, and we hope that Allah grants you what you wish for and rewards you generously for that. There is no doubt that whoever guides others to any righteous deed earns equal rewards to the ones who do that deed. The Prophet, sallallaahu ‘alayhi wa sallam, said, "By Allah, if one man is guided to the right path through you, it would be better for you than the (most precious) red camels." However, this commendable intention does not justify befriending this girl. It is impermissible for the Muslim man to have female friends; Allah, The Exalted, says (what means): {...desiring chastity, not unlawful sexual intercourse or taking (secret) lovers.} [Quran 5:5]
Inviting this girl to Islam does not necessarily entail befriending her. It is permissible for the Muslim man to invite non-Muslim women to Islam provided that he abides by the relevant teachings of Islam; avoiding whatever may lead to Fitnah (temptation), such as being alone with her and the like. It is better in all cases that a woman invites another woman to Islam; this is safer and farther from Fitnah. Therefore, we advise you to seek a righteous Muslim woman who can invite her to Islam, or you may refer her to an Islamic center in that country. You can also give her some Islamic publications and audio materials that explain true Islam. Please, refer to fatwas 88997 and 85379.
If Allah guided her to Islam and you were able to marry her, then this is the desired objective; otherwise, you should cut off your relationship with her and stop thinking about her, for there are many women other than her. You should not be deceived by such online encounters, and the character that this girl displays may be contrary to her real character. You stated that it is socially accepted for her to have a boyfriend in her culture; however, such friendships allow them to commit the worst of sins (adultery) without any social objection to it.
Lastly, we would like to draw your attention to the fact that our compassion and mercy towards you urges us to give you advice, as enjoined by the Sharia. Tameem Ad-Daari narrated that the Prophet, sallallaahu ‘alayhi wa sallam, said, "This religion is founded on the sincere exchange of advice..." Indeed, the one who refrains from advising his fellow Muslims is considered to be deceiving them; Qataadah said, "You would not meet your fellow believer except as an advice-giver, and you would not meet him as a deceiver."
Allah knows best.