You can track that through Google Analytics. You would need to "tag" all the URLs in your emails with tracking parameters using the Google Analytics URL Builder.
Then Google Analytics would show you a campaign report similar to this: (image source)
Here is an article that has best practices for email newsletter tracking with Google Analytics with additional information about how to use the different fields available in the URL builder.
Once you have set up goals, Google will display the number and percentage of users that have completed that goal right in the campaign tracking report.