First, you have to convert single-rail 12V dc power to dual-rail +/-12V dc power and this can be accomplish with the off-the-shelf dc-dc converters ..
For example, if you are happy with 1W dc-dc conversion you can consider something like the NMA1212 ..
Data sheets are here: http://www.seven-segments.com/prjfiles/DS_nma0515d.pdf
Once you have +/- voltage available the second part should not be difficult – you can use a rail-to-rail opamp to produce signal with full +12V-to-12V swing ..