We present the design, implementation, and performance of the digital pulsar observing system constructed for the Canadian Hydrogen Intensity Mapping Experiment (CHIME). Using accelerated computing, this system processes independent, digitally steered beams formed by the CHIME correlator to simultaneously observe up to 10 radio pulsars and transient sources. Each of these independent streams is processed by the CHIME/Pulsar back-end system, which can coherently dedisperse, in real time, up to dispersion measure values of 2500pc/cm^3^. The tracking beams and real-time analysis system are autonomously controlled by a priority-based algorithm that schedules both known sources and positions of interest for observation with observing cadences as rapid as 1 day. Given the distribution of known pulsars and radio-transient sources and the dynamic scheduling, the CHIME/Pulsar system can monitor 400-500 positions once per sidereal day and observe most sources with declinations greater than -20{deg} once every ~4 weeks. We also discuss the extensive science program enabled through the current modes of data acquisition for CHIME/Pulsar that centers on timing and searching experiments.