Fast 5V DC power monitoring
A DSO (Digital Storage Oscilloscope) would be the proper tool to use in order to record the fluctuations of current in your circuit.

While purpose-made oscilloscope current probes are available in the market, they are not inexpensive, and high performance/high precision units can cost thousands of dollars.  This solution may be practical in your situation if you intend to do long-term or frequent/production measurements.

A more cost-effective solution would be to use a precision shunt resistor of a known value in series between the negative supply rail and the ground of your circuit.  You would then measure the voltage across the resistor with the oscilloscope probe, and use Ohm's law to calculate the scale for your oscilloscope.  Many DSOs can even be programmed with custom scaling so you can calibrate your oscilloscope's sample captures.

Which you choose really depends on how much money you want to spend versus how much precision you require.