Measuring uA current with DMM?
It all depends. What accuracy are you looking for? Some DMM have uA range, but the burden voltage is high. You can solve this easily using an external circuit. All you need is a resistor and an precision OpAmp. 

Dave Jones from the EEVBlog designed this circuit that lets you use your DMM in DC Volts mode to measure precisely nA, uA and mA with good resolution. 
You can find the schematic and pcbs designs files in: http://www.eevblog.com/projects/ucurrent/

If you don't want to build the whole circuit it's fine, you can just use the uA part to get a good reading. Just find a suitable resistor and an OpAmp with similar specs. It depends on the bandwidth (how fast the current will be changing).

I have built several of these myself and they work great.

Another solution is to buy some of the Silabs gecko starter kits that include a current measuring circuit (they cost $29USD). It's easy to connect your circuit to those stater kits and via their software you can do several power calculations. You can even see a real time graph.