Let us suppose the family name is 山田.

Most commonly, we would say:

>「[山田家]{やまだけ}」 ← 「家」 is read 「**け**」 for this usage.


>「[山田一家]{やまだいっか}」

***Formally***, we could say:

>「山田[一族]{いちぞく}」

「一族」is not an everyday word.  It could sound too dramatic or theatrical if used in a casual convo.



> "And would you use the same language when talking to a member of the family you're referring to and when talking about that family to a third party?"

The same phrase can be used on both occasions.