Is there a way to have my iPhone 5 report current charging rate?
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