How do I track a custom property for each page on my site? For example, as well as tracking page views, bounce rate, etc for a given page, I want to be able to track a custom property as well (specifically, the number of times someone clicks a specific link on a specific page).
Below is a mockup of what I want to be able to see (either on Google Analytics dashboard or via GAPI):

You can do this with event tracking. See google’s guide for more information:
https://developers.google.com/analytics/devguides/collection/gajs/eventTrackerGuide
Example:
The final param, which I didn’t use is the
non-interactionparameter and it’s usage depends on your particular case and how you would like bounce rate effected. Read about it at https://developers.google.com/analytics/devguides/collection/gajs/eventTrackerGuide#non-interaction