Is there a way to have my iPhone 5 report current charging rate? For example, so I can test different USB ports on my computer and powered hubs to see if one charges my iPhone faster than another.
Since iOS can indicate the charge in percentages, you could simply get a pencil and some graph paper and wake the screen periodically and record the displayed percentages to fit that data to a charging curve.

We have some questions about getting the charge rate programmatically in iTunes, and you could ask if there is an API (private or public) by opening a question on http://stackoverflow.com but for users (as opposed to developers), the answer is no. You can't get a log file of the charge status directly from a computer or the device itself. 

Here are some related questions that may help:


Can I get iPhone charge status from my Mac?
iPhone Status/Overview App Wanted