You actually need to detect the envelope. It's useful to think of this as a demodulation problem, as that's what it is!
Send the current through a small sense resistor that feeds a difference amplifier or instrumentation amplifier. Then, feed that output through a precision full wave rectifier, followed by a low-pass filter.
If you don't rectify first, the low pass filter simply zeros everything out. Rectifying prevents this, giving a non-zero mean that moves with amplitude.
Similar processing is routinely done for electromyography signals (muscle-related electrical signals) which are very AC, but most folks are interested in some average activity. I've seen the rectification/low-pass arrangement called a "Paynter filter" in this particular case.
Filter at whatever corner frequency reliably tracks the changes you're looking for, then sample fast enough to avoid aliasing. Depending on what your corner frequency is, and what filter you use, you might have some phase distortion.
rectifier circuit from sound.westhost.com/appnotes/an001-f4.gif
As an aside, I've found this circuit to be trickier to use than it should be. Perhaps it needs faster diodes than I use, but a small cap across the feedback path of the first op amp seems to do wonders.