Potato peels (PPs) as waste biomass were selected as the biobased carbon source for this study, using urea as N precursor and boron trioxide as B precursor for the “in situ doping” via hydrothermal carbonization (HTC). During HTC, the feedstocks decompose over a wide range of complex chemical degradation mechanisms that finally form single B- and N- as well as B,N-co-doped hydrochars (HCs). Upon chemical ZnCl2 activation, the single B-doped activated carbon (AC) possessed a maximum B content of 0.2 wt%, whereas co-doped B,N-AC had the highest N content of 5.7 wt% with a B content of 0.1 wt%. The influence of single and B,N-co-doping on the physical-chemical material properties of the AC electrodes was analyzed and compared, in combination with its effect on the electrochemical performance for energy storage application. Compared to pristine AC derived from PPs, the B-doped and B,N-co-doped AC depicted increased electrical conductivity (EC) values of 50.3 S ⋅ m−1 and 34.0 S ⋅ m−1, respectively. In addition, the B,N-co-doped AC unveiled the highest average specific capacitances of 51.7 F ⋅ g−1 at 100 mV ⋅ s−1 and of 71.9 F ⋅ g−1 at 5 mV ⋅ s−1 outperforming the specific capacitance values of the reference material AC from peat.