There are two ways you can install premium Magento extensions and each different method has it's own merits.
Before installing any Magento extension, you should disable your Magento Cache and the Magento Compiler. After installing the extension, you should always recompile the Compiler instead of simply enabling it again.
Magento Connect - Downloader
It is possible to install the .tgz extension file automatically through Magento Connect (Downloader). To access your Magento Downloader, add /downloader/index.php to the end of your Magento URL. You can login to the Downloader using your Magento Admin login details.
Once you have logged in, upload the .tgz file using the 'Direct package file upload' form. This will automatically install the extension for you.
Upgrading using the Downloader
If you want to upgrade a premium Magento extension using the Downloader, you must first uninstall the extension. You can do this automatically via the Downloader and then install the new updated version using the steps outlined above.
Installing Magento extensions automatically via the Downloader is the best option. It is much easier and makes it easy to remove an installed extension. That being said, there are many problems with the Downloader that may stop it working for your site (permissions being the main problem). If you are unable to use the Downloader, you can install the extension manually. You can also add the extension to the Downloader so that if in future you fix your Downloader, the extension will be registered there and can be upgraded or uninstalled automatically.
To install the extension manually, extract the files from the .tgz file. Next, ignore the package.xml (this is used by the Downloader to automatically install the extension) and merge the rest of the files with your Magento site using either FTP or SFTP. Refresh the Magento Admin and the extension will be installed.
To register the extension with your Magento Downloader, copy the name of the .tgz file (not including the .tgz part) and rename package.xml as the file you just copied followed by '.xml'. Upload this file to var/package/ and the extension will be registered in your Downloader.
Installing Magento WordPress Integration Add-On Extensions
Add-on extensions for Magento WordPress Integration are just like regular Magento extensions and can be installed using either of the methods outlined above. Before installing though, always ensure that Magento WordPress Integration is upgraded to the latest version. This will reduce the chances of encountering any problems with your new add-on extension.