This style of playing has a few defining characteristics:

a) Frequently changing chords, generally two per measure (in 4/4).
b) A walking bass line beneath the chords.
c) Frequent use of closed-position chords (i.e. with no open strings) instead of open-voiced chords (such as the standard G, C, etc.).
d) "Passing chords" used between the major harmonic areas in a tune.
e) Extended chords (6ths, major 7ths, 9ths, 13ths, minor 7ths, minor 9ths), altered chords (b9ths, augmented, diminished). We want to use these techniques to transform the simple chord progressions that are found in many fiddle tunes and country tunes into more complex progressions that will still fit the melody. This is an important point: what you do still has to fit and make musical sense. You probably won't want to use very complex chords like, say A13b9sus4. It probably won't work well, but just use your judgment. If a chord sounds good then it's good.

These are essentially jazz-based techniques, and it helps to have at least some knowledge of jazz harmony to get a grip on this stuff. The discussion does get a little technical, but even if you don't understand the theory, just try playing through the examples as they are a lot of fun and sound cool.

Listen to recordings by the great Bob Wills and the Texas Playboys for good examples of this type of backup playing. The Texas Playboys guitarist, Eldon Shamblin, is responsible for much of the development of this style as it is applied to country-style tunes. Another great group to listen to if you want to hear this stuff in a more traditional jazz setting is the Quintet of the Hot Club of France, with the virtuoso Django Reinhardt on guitar. Another master practitioner of this style is Ranger Doug from Riders in the Sky.

Here are some chord changes to help you get started

An important point about this backup style is the articulation. You should strive for a sound like this: