How do you begin hiring a lawyer for possible medical malpractice resulting in the death of a familiy member at a hospital?

You should begin by gathering the complete chart from the emergency room visit.  If the family member had visits with other doctors for the same condition (if, for example, he or she had seen his/her primary care doc recently for the same condition) you'll need to gather those records too.  If there were imaging studies done, you'll need copies of the imaging studies themselves, not just the radiology reports.  It is usually helpful if a family member with the best memory writes a timeline setting out what happened and why the family thinks there is a case.  Then, you'll have to find a med mal lawyer in your area.  You can look on the website of the American College of Trial Lawyers (you're looking for a plaintiff's med mal lawyer), the SuperLawyer website, or the Leading Lawyers website--any of these are a decent place to start.  Make sure you pick someone that you like and trust-- if you have a case it will be a long haul and you'll need to be able to work with and trust whomever you choose. I'm sorry for your loss and I wish you the best of luck.