This assumes your current site is running Drupal 7. If you are still using Drupal 6, upgrade the Drupal 7 using the normal upgrade process.
Once a site is already running D7, download the install profile. Replace the sites directory in the install profile w/ your Drupal 7 site's site directory.
cp -R sites ../cm_starterkit_moderate
Using the Devel module or PHP filter in a node, execute...
Using mysql or phpMyAdmin, run
UPDATE system SET status = '1' WHERE name = 'cm_starterkit_moderate';
Remove any modules, themes, and libraries you had added to sites/all that now exist is profiles/cm_starterkit_moderate
If you've already implemented CiviCRM, you'll need to follow the steps for moving CiviCRM to a new location.
If everything is correct, you should see Community Media Starter Kit (Moderate) (cm_starterkit_moderate-7.x-1.xx) in admin/reports/status.
If you only see cm_starterkit_moderate- there is a good chance the install profile has been disabled in the system table for some reason. This will also generate errors in admin/reports/dblog about Notice: Undefined index: name in system_requirements(). To resolve this, reset the status of cm_starterkit_moderate to 1 in the system table and run drush rr.
When configured correctly, the profile will check for bug fixes and security updates along with the other modules. Once you've converted to a kit as the base of your site, use the Steps for Updating the Community Media Starter Kit to update Drupal, CiviCRM, and the modules included in the distribution.