Indeed one method is to measure the current at the anode, except that's not always practical as it requires separate connections for the HV bias and signal. Given that, normally, one would be measure discrete pulses due to individual scintillation events, this can be done at the cathode.
The HV is connected to the PMT via a series resistance that gives a voltage proportional to the PMT current (this will be a sum of steady DC drawn by the bias chain and the photocurrent that flows from cathode to anode (amplified photoelectrons)), because the scintillation events produce small pulses of light, the photocurrent will also be made up of small pulses, these are normally AC coupled via a capacitance from the series resistance to the detector, and is probably how the cdv rate meters operate as the PMT is operating similar to a GM tube.
As far as measuring these voltages, you need a high voltage probe, these are normally made up a string of high value resistors acting as a potential divider, they both attenuate the signal to a measurable level and increase the resistance load on the source. Measuring the pulses should be quite easy if you couple the system to your scope via a high voltage capacitance with a value in the 100nF region