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We present the design, implementation and performance of a digital backend constructed for the Canadian Hydrogen Intensity Mapping Experiment (CHIME) that uses accelerated computing to observe radio pulsars and transient radio sources. When operating, the CHIME correlator outputs 10 independent streams of beamformed data for the CHIME/Pulsar backend that digitally track specified celestial positions. Each of these independent streams are processed by the CHIME/Pulsar backend system which can coherently dedisperse, in real-time, up to dispersion measure values of 2500 pc/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 small as one day. Given the distribution of known pulsars and radio-transient sources, the CHIME/Pulsar system can monitor up to 900 positions once per sidereal day and observe all sources with declinations greater than $-20^\circ$ once every $\sim$2 weeks. We also discuss the science program enabled through the current modes of data acquisition for CHIME/Pulsar that centers on timing and searching experiments.