Problem with the AF-C on my Nex-5R (shakes like crazy)

It's normal behavior.  It's the contrast auto focus detection pumping at the extremes of focus to try and ensure the shot it takes is in focus.  I forget if this occurs with the 5r or the 6 as I know this happens with the 5n but AF-C is also the only way to get the camera to adjust exposure every shot in any burst mode.