How likely am I to become infected with hepatitis C from a family member?

It depends. Depends on what you are doing with this family member. Sharing needles (drugs, injected vitamins or other home remedies) is high risk. Kissing, hugging, sharing dishes is no risk. Sharing personal items that may contain small amounts of blood like toothbrushes, razors, or nail clippers does have risk.
Unlikely. Hepatitis c is spread by exposure to blood typically. So one wouldn't want exposure to an open cut or to share a toothbrush or razor. This aside, shaking hands or other casual contact is felt to be safe.