What do I need to know to claim what's rightfully mine and do I need a paternity test to prove that he is my father?

The question is was he ever declared to be your legal father in a court of law. This would be the case if there was a child support order or a parenting plan or some a custody order. In the absence of any such legal determination, some states allow a post mortem paternity test. All of that assumes of course, that he did not leave a will, name you as his child and then leave you nothing. If he did, then you get nothing as all that is required is that he name you.