We use depth- and time-resolved x-ray diffraction to study thermal transport across single crystal Bi films grown on sapphire in order to determine the thermal conductivity of the film and the Kapitza conductance of the interface. Ultrafast Ti:sapphire laser pulses were used to heat the films; x-ray diffraction then measured the film's lattice expansion. Use of grazing incidence diffraction geometry provided depth sensitivity, as the x-ray angle of incidence was varied near the critical angle. The shift of the film's Bragg peak position with time was used to determine the film temperature averaged over an x-ray penetration depth that could be selected by choice of the angle of incidence. For films that were thick compared to the laser penetration depth, we observed a large temperature gradient at early times. In this case, measurements with the incident angle near or well above the critical angle were more sensitive to the film conductivity or Kapitza conductance, respectively. For thinner films, however, cooling was dominated by the Kapitza conductance at all accessible time scales.