First of all, If I sound too lecturing, please let me know. I'm a picky person, but I try to be aware of it.
First of all, DOF is depth of field. Which field?. The "acceptable focus" field. That is how deep from the in focus plane you will achieve acceptable focus. You may know that, but that's why you don't get "dof of background", because there is not such a thing since you cannot get the background "depth of fielded".
Depth of field is controlled by the diaphragm, whose aperture is represented by the f/ number. The larger the aperture, the shallower of your depth of field; the smaller your aperture gets, the deeper your acceptable focus field is.
Now, the larger your aperture, the smaller your f/ number will be. If you, as you said, have a camera that goes from f/2.3 to f/8 your largest aperture is f/2.3 and your smaller is f/8. So, in your camera, f/2.8 will give you the shallowest depth of field you can achieve.
Unfortunately, the effect varies between cameras whose recording medium (film, sensor, daguerrotype or whatever) is different: the smaller your sensor (or recording medium), the deeper your DOF will be at the same aperture. This means that your f/2.3 could be (very roughly) equivalent to f/11 in a 35mm camera (or full frame sensor camera).
You may see pictures taken with DSLRs at f/8, for example, that have a much shallower DOF than yours with f/2.3.
Here is a comparison between my Canon A70 and my EOS 400D.
This photo was taken @ f/2.8 with my A70, in macro mode (which makes DOF shallower) and
very close to the subject.
This one was taken @ f/8 (your minimum aperture) with my 400D not particularly close to the subject:
What you can do to have the shallowest DOF possible is to enable macro and get as close as you can. In order to control what gets in focus and what doesn't, you can disable AiAF (this will allow your to focus only with the central AF zone), focus on your subject and recompose the scene if applicable. By using macro and get really close, you will be able to get blurry backgrounds (and foregrounds, if something is closer to the camera than the focal plane).
Unfortunately, it may be the camera much more than the photographer.