Yes, you can make the armature match the position of a bezier curve with a spline IK constraint. Setting it up is a lot like making an IK constraint but with a couple extra steps.
1) Setup your bones. It looks like you have done that already, though be warned: since it is an IK constraint it works from the tip of the hierarchy towards the Root, not the other way around.
2) Add a bezier curve and add the number of control points you want in the form of edge vertices.
3) Add some control objects for the curve. Many people just add some empties for this purpose, but there is nothing stopping you from just making armature bones. However: it is not a good idea to use bones from the Armature that is being controlled by the Spline IK, as it will likely create a cyclical dependency.
4) Hook those vertices of the curve to the control objects. (go into the curve's edit mode, click a vertex, ctrl+H, choose appropriate option).
5) Apply the Spline IK constraint to the last bone of the chain. Configure "Chain Length" to match the number of bones in the curvy part of the armature. Play with the other settings as you see fit.
When done, you should be left with a curvy poseable armature controlled with your control objects. You can mess with the control handles of the curve by rotating and scaling your control objects.
The rig shown here is a bit more complicated, since I did want bones to control how this armature worked. It involves creating two armatures, one parented to another, and the top level armature controls everything (including the Bezier curve).