Reconfigurable intelligent surface (RIS) is a promising technology for 6G networks owing to its superior ability to enhance the capacity and coverage of wireless communications by smartly creating a favorable propagation environment. In this paper, we investigate the potential of employing RIS in dual-functional radar-communication (DFRC) systems for improving both radar sensing and communication functionalities. In particular, we consider a RIS-assisted DFRC system in which the multi-antenna base station (BS) simultaneously performs both multi-input multi-output (MIMO) radar sensing and multi-user multi-input single-output (MU-MISO) communications using the same hardware platform. We aim to jointly design the dual-functional transmit waveform and the passive beamforming of RIS to maximize the radar output signal-to-interference-plus-noise ratio (SINR) achieved by space-time adaptive processing (STAP), while satisfying the communication quality-of-service (QoS) requirement under one of three metrics, the constant-modulus constraint on the transmit waveform, and the unit-modulus constraint of RIS reflecting coefficients. An efficient algorithm framework based on the alternative direction method of multipliers (ADMM) and majorization-minimization (MM) methods is developed to solve the complicated non-convex optimization problem. Simulation results verify the advancement of the proposed RIS-assisted DRFC scheme and the effectiveness of the developed ADMM-MM-based joint transmit waveform and passive beamforming design algorithm.