Jesus went to great lengths to convince us that family is never ultimate. For example, he put little trust in his biological family. After his ministry began his own hometown rejected him (Luke 4:18-32).
When his family approached him expecting special attention, he made it clear that the family he first identified with were those who heard the word of God and performed it (Luke 8:19-22).
Jesus even told us to hate our families (Luke 14:26) "If anyone comes to me and does not hate his own father and mother and wife and children and brothers and sisters, yes, and even his own life, he cannot be my disciple." Of course by hatred he meant, "Love God so much that by comparison your love for your family is hatred."
He warned us that some members of our family would be the first to turn on us and persecute us even unto death (Luke 21:16–17) "You will be delivered up even by parents and brothers and relatives and friends, and some of you they will put to death. You will be hated by all for my name’s sake. "
For someone who loves their family, this is a problem. I love my family, especially my wife and children. So, how do I interpret this? What is the solution? The solution is a call to a deeper love for God, and dependence upon God not family. It is also a call to a healthy distrust of sin first in myself, and second in those closest to me. Last, it is a call to gratitude for faithful, loving, Christ-centered family members. This is a luxury none of us deserve.