Millimeter wave (mmWave) radar technology has potential applications in vital signs detection and medicine. In order to minimize the influence of human micro-movements and respiratory harmonics on heart rate estimation, the vital signs detection method based on mmWave radar is studied in this paper. First, we use median filtering to eliminate baseline drift caused by human micromotion. Next, a differential recursive least squares multiple classification (DR-MUSIC) algorithm is proposed based on the combination of recursive least squares-based adaptive filter (RLS) and multiple signal classification (MUSIC) algorithm. This algorithm effectively suppresses respiratory harmonics and separates respiratory and heartbeat signals. Finally, heart rate value can be precisely estimated using spectral peak search. We invite a number of people to participate in the experiment, which demonstrate that the method successfully suppresse the impact of respiratory harmonics at low SNR. The error rate between the estimated heart rate and the reference heart rate is only 1.69% to 2.61%, which is significantly better than the existing algorithms.