When is it permissable to end a relationship with a family member? What would be the circumstances under which this is permitted by the torah and commentaries?
The question is a little vague. I understand that you are not referring to a spouse, but to another member of the family.
The Torah does not permit one Jew to hate another, as the verse writes, “you shall not hate your brother in your heart.” The Sages write that intentionally refraining from speaking with another for three days is a sign of hatred, and this would be prohibited. Only if the other person would be classified as “wicked” would it be permitted to hate him.
If the person has hurt or injured you, the best advice is to speak with him, and hash it out with him (see Chovas Halevavos, Nekimah 15, Derech Mitzvosecha 17; Rabbeinu Yonah on Mishlah 10:18; Or Hachaim Vayikra 19). There is no obligation, however, to be “best friends” with everybody, however, with family members it is especically important to try to make up, for family rifts can spread to others, and the Torah writes “you shall not ignore your flesh” — your family members.