The first signal model for a single photon avalanche diode (SPAD)-array communication receiver for multilevel modulation schemes is reported. This paper proposes a novel, generalised SPAD array signal and noise model for both digital and analogue, synchronous and asynchronous SPAD readout arrays, which includes the competition between the input photons, dark counts and after-pulsing counts. With this contention signal and noise model, multilevel signals including the signal variation after distortion or equalisation can be evaluated. Also, we report the first numerical investigation for SPAD-based, high data rate, free space, visible light communication using higher order pulse amplitude modulation (PAM) with matched filter, linear and non-linear Volterra post-equalization. Simulations have been carried out to analyze and compare the bit error rate (BER) performances under a variety of conditions. The model is verified by comparison with published experimental results.