If you're using Google Chrome Extensions Gallery to publish your extensions, Google added "support for Google Analytics, which can help you better understand how many people visit your extension pages, where they're coming from and more. Starting today, you can specify a Analytics profile for each of your extensions. This'll start tracking the extension's page in your Google Analytics account as if that page was part of your own site," informs Google.
You can also use Google Analytics to track the usage of your extension once it's installed. Check this Tutorial that explains how to integrate Analytics such that you can analyze how users interact with features of your extensions.
More Info: Setup instructions