A quick and dirty method is with a DSO and a 1 ohm resistor in power lead.
Set DSO up for infinite persistance, and look at V drop across R, that will give you peak.
Now set DSO up to average its input channels, and look again at drop across R.
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If you happen to have a multimeter with logging capability, use the approach with the 1Ohm resistor (or a larger one if you maximum current is lower than say 100mA). You can then use a spreadsheet (or whatever software the multimeter brings with it) to calculate what you need.