Great answer here from DeReel.
I think the biggest challenge with doing this with a family is that you are going to have a very different level of interest and involvement from each person (as opposed to a gaming group, which has gathered with the explicit intent to engage in this particular activity).
Some people get really excited about roleplaying, or really invested in the characters or the adventure, and other people just don’t want to engage. So, you’ll have to be prepared for that, whether it just means to be emotionally prepared, or to have a backup plan for those who are less interested.
What kind of people constitute your family? Is this you and your spouse and children, or something more unusual? Any teenagers? Anyone with RPG experience? Are they competitive, shy, entertainers?
You could bring in someone with RPG experience to run a game for you, as well. That might change the dynamic.
As for game and genre, try to figure out what your family is into - favourite novels, films, TV shows, art projects? It would be good to start with something everyone can get excited about.