You can adjust the "All" tab to only view the "Family" circle (or any other set of circles you want).
To do this:
View the stream of each other circle by selecting that circle.
In the upper right hand corner above the stream, there is a slider that selects the volume. Set the volume all the way down (to the left).
Repeat this for each of the circles you do NOT want in your main stream.
For your "Family" circle, move the volume all the way up (to the right) to make sure you get all of the posts from that circle.
The volume slider only controls what shows up in the main stream - you can still view all the posts in each circle by visiting that circle.