To measure the current you need something like a modified micro USB cable which is sliced open and the +5V line (normally red) split so that you can insert the multi-meter on the current range (e.g. 2A) between the two halves of the split. You can make a more general purpose 'break-out' cable out of a short USB A male to A female cable which is split in the same way and can then be inserted in between the power supply and the normal power lead. This can then be used to measure the current flow through most USB connections.
If you do this a lot then it is quite useful to make this into a real project breakout box with a switch and other monitor points like in the image. The switch opens / closes the +5V line between the two sides; the blue socket is the 0V line, the monitor points are the two sides of the +5V line and the 2 data lines (only needed if one is going to start scoping the data). The monitor points are from a recycled Molex disk drive power connector which I find a good 'socket' for multimeter leads. The voltage on the cable can be measured across the 0V and either of the +5V lines with the switch closed. The current can be monitored by putting the meter in current mode between the +5V lines and then opening the switch. If you have two meters then you can measure both current and voltage together and see if the power supply drops when you put extra load on.
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