This is a great question, and as a long time teacher of The Adventures of Huckleberry Finn, one I haven't thought about. But, here goes . . .
Let's think about some of the families in the novel and what they could all have in common. There is Huck's family, the Grangerfords and Shepardson's, Jim's family, the Wilks family, and Tom's extended family with Aunt Polly.
Probably the most important "family" that exists in the book are not blood relatives--it is the family that Huck and Jim form in their adventures together.
Here are some short explanations of the families--
Huck's--Huck's mother is dead, and he has an alcoholic father whose only interest in Huck occurs when he learns that Huck is rich. He is physically abusive to Huck. He kidnaps Huck and even attempts to kill Huck before Huck fakes his own death and escapes to Jackson Island. Pap is a racist, a gambler, and low life who doesn't care about the well-being of his own son. The fact that Huck turns out with a conscience is an indication that one can overcome things despite his upbringing.
Grangerfords and Shepardsons--Both families are rich and own slaves. They are large families who will sacrifice their own kin to a feud that no one even remembers why it started. They are ruthless murderers who put pride and honor over the lives of others.
Jim's--Jim's family has been sold away from him, and he desperately wants to buy his family back. Although he hit his daughter once, he is deeply sorry for his actions (unlike Pap who abused Huck). To him, family is important, and one of the reasons why he feels love for Huck in the form of protecting him and guiding him down the river like a real father would.
Wilks--Probably more of a traditional family structure; however, the Wilks girls are alone in the world and fall prey to the King and Duke who try to steal their inheritance. They are also slave owners who have no real sympathy for their slaves. They will sell them willingly to gain back the money they thought was stolen from them by the King and Duke.
Tom's--Although we only really meet Aunt Polly, Tom's family are slave owners as well. They, however, seem to be more concerned about their slaves especially when Jim saves Tom's life. However, they didn't have any problem locking Jim up in a shed when they thought he was a runaway slave and could get a reward for him. Tom's family shows hypocrisy on many levels.
If you put this all together, Twain's opinion about family is a little skewed even though he had a very happy life with his wife and children. His most important point is that you don't have to be related to be family. All of the other families he writes about in the novel are flawed in one way or another, but no family is perfect.
So, maybe that's the attitude or the tone he wishes to present in the novel. Disgust for family is normal; families do disgusting things sometimes, and you have to make your own family anywhere you can.