We present an electrochemical exfoliation method to produce controlled thickness graphene flakes by ultrasound assistance. Bilayer graphene flakes are dominant in the final product by using sonication during the electrochemical exfoliation process, while without sonication the product contains a larger percentage of four-layer graphene flakes. Graphene sheets prepared by using the two procedures are processed into films to measure their respective sheet resistance and optical transmittance. Solid-state electrolyte supercapacitors are made using the two types of graphene films. Our study reveals that films with a higher content of multilayer graphene flakes are more conductive, and their resistance is more easily reduced by thermal annealing, making them suitable as transparent conducting films. The film with higher content of bilayer graphene flakes shows instead higher capacitance when used as electrode in a supercapacitor.