High surface area electrode materials are of interest for a wide range of potential applications such as super-capacitors and electrochemical cells. This paper describes a fabrication method of three-dimensional (3D) graphene conformally coated on nanoporous insulating substrate with uniform nanopore size. 3D graphene films were formed by controlled graphitization of diamond-like amorphous carbon precursor films, deposited by plasma-enhanced chemical vapour deposition (PECVD). Plasma-assisted graphitization was found to produce better quality graphene than a simple thermal graphitization process. The resulting 3D graphene/amorphous carbon/alumina structure has a very high surface area, good electrical conductivity and exhibits excellent chemically stability, providing a good material platform for electrochemical applications. Consequently very large electrochemical capacitance values, as high as 2.1 mF for a sample of 10 mm(3), were achieved. The electrochemical capacitance of the material exhibits a dependence on bias voltage, a phenomenon observed by other groups when studying graphene quantum capacitance. The plasma-assisted graphitization, which dominates the graphitization process, is analyzed and discussed in detail.