Obtain a high performance current measuring instrument. Use digital multimeters that offer current ranges in the order of a billionth of an ampere, also known as the nanoampere range.
Optimize your measurement setup. Use measurement probes and clips that have a low resistance in relation to the resistance in your circuit. Avoid long wires in the measurement setup. Connect your probes as close as possible to the point where the current exits the circuit and where it reenters.
Model your setup. Find out the resistance of the leads that connect to your circuit and compare them to the resistance in your circuit path. Use low resistance leads that have a resistance that is at least 10 times less than the resistance of the path through which you are measuring current flow. Use leads with a resistance of less than 100 times the resistance of the path if you want a measurement that is highly accurate.
Calculate an estimate of the current. Use your electronics model to determine what the current should measure when the ammeter meter leads are not connected and when they are connected. Use the resistance of the leads to calculate the error that the leads will introduce.