Is it permissible to marry a girl from a difficult family background, and how can one convince the family to accept this marriage, taking into account their concerns about "lineage being ingrained" and the possibility of the girl being righteous despite her family circumstances?

The Sharia (definition) does not permit love relationships between young men and women. There is no objection to marrying a girl who was in a love relationship, provided she becomes righteous. A righteous woman is not held accountable for the corruption of her family, for Allah Almighty said: "And no bearer of burdens will bear the burden of another." The hadith (definition) "The lineage is insidious" is weak, and even if authentic, it refers to considering the religion and character of the woman herself. If your father refuses your marriage to her, convince him or seek mediation from someone who can influence him. If he insists on refusing, then obey him.