To measure the charging behavior of a Cockcroft-Walton multiplier, it's essential to observe the voltage over time without discharging the capacitors, which can be done using an oscilloscope with minimal loading. A differential input oscilloscope may help visualize the waveforms throughout the circuit. The user is currently unable to see the expected amplitude increase in their half-wave multi-stage setup, suggesting potential issues with the driving frequency. It's debated whether to measure voltage across the highest capacitor stage or relative to ground, with the latter being necessary if the driving source is not floating. Observing transient waveforms effectively may require a storage oscilloscope or synchronized camera for accurate results.