This paper proposes an interleaved transceiver design method for a multiple-input multiple-output (MIMO) integrated sensing and communication (ISAC) system utilizing orthogonal frequency division multiplexing (OFDM) waveforms. We consider a continuous transmission system and focus on the design of the transmission signal and a receiving filter in the time domain for an interleaved transmission architecture. For communication performance, constructive interference (CI) is integrated into the optimization problem. For radar sensing performance, the integrated mainlobe-to-sidelobe ratio (IMSR) of the beampattern is considered to ensure desirable directivity. Additionally, we tackle the challenges of inter-block interference and eliminate the spurious peaks, which are crucial for accurate target detection. Regarding the hardware implementation aspect, the power of each time sample is constrained to manage the peak-to-average power ratio (PAPR). The design problem is addressed using an alternating optimization (AO) framework, with the subproblem for transmitted waveform design being solved via the successive convex approximation (SCA) method. To further enhance computational efficiency, the alternate direction penalty method (ADPM) is employed to solve the subproblems within the SCA iterations. The convergence of ADPM is established, with convergence of the case of more than two auxiliary variables being established for the first time. Numerical simulations validate the effectiveness of our transceiver design in achieving desirable performance in both radar sensing and communication, with the fast algorithm achieving comparable performance with greater computational efficiency.