Is it permissible to marry a Muslim woman who has lost her virginity due to an illicit relationship, knowing that she has repented and wishes to become righteous, and does her past affect the possibility of marrying her?

Your acquaintance with this girl and establishing a relationship with her is not permissible. The only permissible relationship between the sexes is marriage. The girl should have repented and concealed herself, as Islamic law prohibits traveling without a mahram (unmarriageable kin) and seclusion with non-mahrams, and zina (fornication/adultery) is a grave sin. However, Allah has opened the door to repentance from it. What happened is not an impediment to you marrying her if she repents and you verify that her womb is clear, for "the one who repents from sin is like one who has no sin." It is permissible to marry a woman who has committed zina if two conditions are met: her repentance to Allah and her purification (from any potential pregnancy). You should seek out a righteous woman of faith.