© 2018 The Hamamatsu R5912-HQE photomultiplier-tube (PMT) is a novel high-quantum efficiency PMT. It is currently used in the DEAP-3600 dark matter detector and is of significant interest for future dark matter and neutrino experiments where high signal yields are needed. We report on the methods developed for in-situ characterization and monitoring of DEAP's 255 R5912-HQE PMTs. This includes a detailed discussion of typical measured single-photoelectron charge distributions, correlated noise (afterpulsing), dark noise, double, and late pulsing characteristics. The characterization is performed during the detector commissioning phase using laser light injected through a light diffusing sphere and during normal detector operation using LED light injected through optical fibres.