The dynamics of laser-induced plasma/shock wave and the interaction with a surface in the laser shock cleaning process are analyzed by optical diagnostics. Shock wave is generated by a Q-switched Nd:YAG laser in air or with N$_2$, Ar, and He injection into the focal spot. The shock speed is measured by monitoring the photoacoustic probe-beam deflection signal under different conditions. In addition, nanosecond time-resolved images of shock wave propagation and interaction with the substrate are obtained by the laser-flash shadowgraphy. The results reveal the effect of various operation parameters of the laser shock cleaning process on shock wave intensity, energy-conversion efficiency, and flow characteristics. Discussions are made on the cleaning mechanisms based on the experimental observations.