If you have registered your installation with the purchased license key, all updates become available directly into your Dashboard. In addition to the regular update messages of WordPress, also the Brooklyn Theme Option inform you about possible updates.
Before you move on, please make sure you have installed the following plugin: http://wordpress.org/plugins/easy-theme-and-plugin-upgrades/
If you are using a child theme. You need to compare the changes we have made with the custom file you got. If you are using windows we can recommend a tool like: http://winmerge.org/ To see which files we have changed with the update please open the changelog.txt inside the brooklyn folder. We have added a note which makes it easier to find changes in files.