Is there a cross platform C API that can be used to get system usage metrics?
Unfortunately not.
The C standard is pretty much limited to dynamic allocation, string manipulation, math and text I/O. Once you get beyond that, you need OS APIs which by definition are OS specific and not cross platform.
I've worked with libstatgrab before. Gets you some pretty useful system statistics for the main Unix-like variants and Windows through Cygwin (supposedly - never tried). Different OS's work so differently - especially when it comes to usage metrics - it may be challenging to get what you want. Even something as simple sounding as "free memory" can be tricky to act on in a cross-platform way. Perhaps if you narrow things down a bit, maybe we can find something.
Depending on the metrics you want to collect, you may want to consider looking at PCP (Performance Co-Pilot). This is an open-source performance framework, originally developed at Silicon Graphics, which collects and collates a vast number of possible metrics from a vast number of sources, and lets you monitor them from anywhere.
Basically PCP would involve adding another 'layer' into your system -- for example, you might monitor a distributed cluster of mixed-OS machines, each with PCP installed locally; a set of 'agents' collect the performance data on each machine, and your code could then use libpcp to collect those metrics as required.
It's hard to say without knowing your exact usage scenario (if you're talking about something running seamlessly on end-users' machines, PCP may not suit but if you want to monitor machines which you control, and are happy to run the PCP service on them, it's an awesome solution).
We use PCP very happily to collect metrics from Windows and Linux boxes, as well as internal metrics from our application, and log them all centrally, report on them, monitor trends, etc.