What is the best way to measure and record the power consumption of a servo motor accurately?
measure the power consumption of a servo motor when it is turning

For this you need to measure the voltage (U) and the amperage (I) at the same time and over time.
That will give you the power consumption (P = U x I)
But the power consumption will vary depending on what phase the servo is in, start, stop and different loads.
So the best way is to log the measurements over time so you can get a accurate overview over the power consumption. This can be achieved with a multi meter with data output, a oscilloscope with logging function or a DIY Arduino (example http://www.electronoobs.com/eng_arduino_tut84.php)
The best solution depends on what you need to measure (mAmps or Amps) and on your wallet and knowledge.
PS. Next time add more data and you will get a more accurate answer.