A one-dimensional Vlasov-Poisson simulation code is employed to investigate the plasma sheath considering electron-induced secondary electron emission (SEE) and backscattering. The SEE coefficient is commonly treated as constant in a range of plasma simulations; here, an improved SEE model of a charged dielectric wall is constructed, which includes the wall charging effect on the SEE coefficient and the energy dependency of the SEE coefficient. Pertinent algorithms to implement the previously mentioned SEE model in plasma simulation are studied in detail. It is found that the SEE coefficient increases with the amount of negative wall charges, which in turn reduces the emissive sheath potential. With an energy-dependent SEE coefficient, the sheath potential is a nonlinear function of the plasma electron temperature, as opposed to the linear relation predicted by the classic emissive sheath theory. Simulation combining both wall-charging effect and SEE coefficient' energy dependency suggests that the space-charged limited sheath is formed at high plasma electron temperature levels, where both sheath potential and surface charging saturate. Additionally, different algorithms to implement the backscattering in the kinetic simulation are tested and compared. Converting backscattered electrons to secondary electrons via an effective SEE coefficient barely affects the sheath properties. The simulation results are shown to be commensurate with the upgraded sheath theory predictions.