Using an updated population synthesis code, we study the formation and evolution of black holes (BHs) in young star clusters following a massive starburst. This study continues and improves on the initial work described by Belczynski and coworkers. In our new calculations we account for the possible ejections of BHs and their progenitors from clusters because of natal kicks imparted by supernovae and recoil following binary disruptions. The results indicate that the properties of both retained BHs in clusters and ejected BHs (forming a field population) depend sensitively on the depth of the cluster potential. In particular, most BHs ejected from binaries are also ejected from clusters with central escape speeds Vesc<~100 km s-1. Conversely, most BHs remaining in binaries are retained by clusters with Vesc>~50 km s-1. BHs from single-star evolution are also affected significantly: about half of the BHs originating from primordial single stars are ejected from clusters with Vesc<~50 km s-1. Our results lay a foundation for theoretical studies of the formation of BH X-ray binaries in and around star clusters, including possible ``ultraluminous'' sources, as well as merging BH-BH binaries that will be detectable with future gravitational wave observatories.