Is it permissible to marry a righteous girl whom the family rejects due to her father's bad reputation, and because she is in need of help due to her harsh family circumstances?

One must choose a wife who is religious, of good character, and from a good family. One should not be led by emotion without thinking. If you are determined to marry a girl whose family is not righteous, try to convince your parents, keeping in mind that obedience to parents is obligatory unless you fear falling into fornication.

Parents should not object to their son marrying a righteous girl if his heart is attached to her, especially if there is kindness to the girl in that.

One must also be careful not to fall into forbidden acts such as mixing (of genders), seclusion (with a non-mahram), and looking at a non-mahram woman. Consult experts and perform Istikhara (prayer for guidance) before making any decision.