Joint communications and sensing (JCAS) systems have recently emerged as a promising technology to utilize the scarce spectrum in wireless networks and to reuse the same hardware to save infrastructure costs. In practical JCAS systems, dual functional constant-modulus waveforms can be employed to avoid signal distortion in nonlinear power amplifiers. However, the designs of such waveforms are very challenging due to the nonconvex constant-modulus constraint. The conventional branch-and-bound (BnB) method can achieve optimal solution but at the cost of exponential complexity and long run time. In this paper, we propose an efficient deep unfolding method for the constant-modulus waveform design in a multiuser multiple-input multiple-output (MIMO) JCAS system. The deep unfolding model has a sparsely-connected structure and is trained in an unsupervised fashion. It achieves good communications-sensing performance tradeoff while maintaining low computational complexity and low run time. Specifically, our numerical results show that the proposed deep unfolding scheme achieves a similar achievable rate compared to the conventional BnB method with 30 times faster execution time.