I think this is an awesome idea. I don't really play videogames very much, if at all lately, but I would definitely pick this up if it was done right.
Playing as Barnabas and Barnabas only would be interesting, but might get boring, especially in certain situations where someone else would be better suited. But truth be told, I wouldn't be happy if they went the route of "You play Barnabas when stalking a victim through the woods, but you also get to play Julia when Barnabas gets trapped in another time band, and you get to be Quentin when Count Petofi shows up to take over..." basically the whole changing characters depending on what situation you are in deal. Buffy: Chaos Bleeds is like that, where you HAVE to be a certain character depending on a level. Maybe if you could select which character you want to be before you start the level, that would be different.
I like the intial idea about you being the descendant of the Collins family, but maybe that could be twisted and supported with some of the original show ideas. For example:
You recieve a letter one day telling you that you are the last original member of the Collins family, a family you've never even heard of. You get a package with this letter that seemingly proves that all of this is true, so you set off for the small fishing village of Collinsport, Maine. When you arrive, you step off the train and head to the Collinsport Inn, where some old crazy lady, wearing a blonde wig (snigger) tells you quick wittedly that you are a jerk, J-E-R-K, jerk! if you go up to Collinwood to search for your lineage. But you decide to go anyway. Once you get up to the estate, you find it in disrepair, but being the adventerous type, you decide to create a makeshift living arrangement out of still well-standing room in the West Wing, that has a large red gramophone with flowers painted on it. You find various clues that lead you to things that help to unlock the history of your life (the Collins Family History, Julia's diary, an old menu of Mrs. Johnson's, some of Roger's business papers, etc.).
Then one night, a mysterious fire erupts from a house of cards you've built while trying to fall asleep, and your room is ruined, so you find another room in the East Wing, one with a mysterious portrait of a blonde woman hanging above the mantle.
This game would work well as a "Choose Your Destiny" type of thing. Like you could go to that room in the East Wing OR you could see the Old House from the hallway window of Collinwood, an decide to check it out, only to find the portrait of a blonde haired woman in a rust colored servants gown above the mantle.
(I wrote a lot more for both of these scenarios, but decided to cut it from this post because I thought it would be breaking the fan fiction rules for the board. If you're interested, PM me and I'll send them to you. Choose your destiny. PT room portrait, or Old House portrait?)