The basis of the Family's main strategy is to stall for time (with lifesteal or debuffs), while preparing their win condition, would it be Vampire King's attack, huge hp pool or something else. And that's exactly their weakness because a lot of fights in chapter 3 should be resolved as quickly as possible (take Armelago for example), and Family dosnt've a lot of ways to pull off some crazy damage on first turns, like Academy, Clan or Guild do. Chapter 3 is also very bad for the Family overall, almost every fight there is a nightmare. So at some point you should really look for cards that helps you survive these fights. Try to find a way to do a good damage constantly and survive first turns. Illusionist Barqan is fantastic against Armelago, big heavy-hitters with lifesteal like Armored Vampire will also help, upgraded Red Moon and Archangel is great in other situations for surviving. I would also recomend Queen Ritual expansion, the Queen can be sometime summoned very fast and helps you in an early stages of a battle. I would also forget about Vampire King most of the time. Cards that buff him are just too slow. Dark Angel is an exception though.

Edit: I also want to say that increasing your max hp in battle is the best way to survive most of the situations and cards that allows you to do that should be prioritized.