A critical element in the realization of large liquid and gas time projection chambers (TPCs) is the delivery and distribution of high voltages into and around the detector. Such experiments require of order tens of kilovolts to enable electron drift over meter-scale distances. This paper describes the design and operation of the cathode feedthrough and high voltage distribution through the field cage of the NEXT-100 experiment, an underground TPC that will search for neutrinoless double beta decay $0\nu\beta\beta$. The feedthrough has been demonstrated to hold pressures up to 20~bar and sustain voltages as high as -65~kV, and the TPC is operating stably at its design high voltages. The system has been realized within the constraints of a stringent radiopurity budget and is now being used to execute a suite of sensitive double beta decay analyses.