Sliding friction between a silicon tip and a polymer bilayer system consisting of a polystyrene (PS) film covered with a few-nanometers-thick capping layer of hard plasma polymer is studied using friction force microscopy. The system was chosen to enable subsurface dissipation channels to be distinguished from surface friction. Frictional energy dissipation in the underlayer can be identified through the kinetics of the polymer relaxation modes that we measured using nanoscale friction experiments as a function of sample temperature, scanning velocity, and applied load. We found a strong nonlinear increase in friction as a function of applied load around the glass-transition temperature of the PS underlayer. This behavior is a clear signature of frictional dissipation occurring in the volume of the polystyrene layer, well below the surface of the sample. The time-temperature kinetics associated with frictional energy dissipation into the PS was found to be in agreement with the known material properties of PS. Moreover, the data was found to support the hypothesis that the observed friction can be understood as the sum of friction resulting from the relaxation process in the polymer underlayer induced by stress due to the sliding of the tip and a second term associated with dissipation due to sliding friction on the capping layer.