 Electrochemical and energy storage properties of thin layer hybrid supercapacitors in the solid-state platform utilizing two asymmetric poly(3,4-ethylenedioxythiophene) PEDOT and graphene electrodes with 1-butyl-3-methylimidazolium tetrafluoroborate ([BMIM][BF4]) ionic liquid gel electrolyte are described. In hybrid design, energy storage is by electrical double-layer charges at graphene and through reversible faradic redox process at PEDOT while the high operational voltage of 2.7 V due to ionic liquid electrolyte boosted the energy density. The PEDOT film was synthesized by pulsed current electropolymerization as confirmed by Raman analysis and was in the microporous form for pervasive access to electrolyte ions. Areal mass of PEDOT was varied and the hybrid supercapacitors with PEDOT/graphene active mass ratio 0.35, 0.46, and 0.78 were analyzed for specific capacity and charge–discharge behavior in order to balance the charge and charge transfer kinetics for optimized hybrid supercapacitor device. Randles–Sevcik analysis showed high ClO4− ion diffusivity 6.6 × 10−9 cm2 s−1 at 2.7 V in ionic liquid gel which is comparable to liquid electrolytes. By combining the micro-porosity of PEDOT, large 600 m2 g−1 surface area of graphene and high 2.7 V stability of ([BMIM][BF4]) ionic liquid gel electrolyte, energy density of 14.9 Wh kg−1 at specific power rating of 9.8 kW kg−1 are realized and stability over 2000 charge–discharge cycles is shown. Impedance and Bode analysis using equivalent circuit model is presented. The characteristics of solar electricity storage are described which can have applications as autonomous energy source harvesting light energy for powering portable power electronics.