First of all, you'll need the SVN version of Kickstart which supports plain file or FTP file access modes, integrates AJAX and JavaScript redirects modes and includes a few more improvements. Since this is not released yet, you can download a developer's preview - but very stable already! - version from here. Extract the PHP file in the archive you've just downloaded.
Next up, you'll need the ZIP version of the Joomla! patch, which you can download from the Joomla! project's file repository site. Depending on which Joomla! version you've already installed, download the appropriate ZIP file. The tar.gz file will not do! For example, if you've got Joomla! 1.5.6 and want to update to 1.5.7 you must download the Joomla_1.5.6_to_1.5.7-Stable-Patch_Package.zip.
Upload the Joomla! ZIP file and kickstart.php to your site's root. If you used FTP, make sure both files' permissions are 0777. Make sure you have a working backup of your site before you proceed! In the unlikely event you run into a problem we want to be sure you can roll back. There are many ways to take a backup of your site, but we obviously suggest you to use the open source JoomlaPack backup component.
Now visit the kickstart.php URL, which is something like http://www.example.com/kickstart.php, substituting www.example.com with the full URL to your Joomla! site. In the page presented to you, make sure the update ZIP is selected in the topmost combo box. For the operation method we suggest using the "JavaScript Redirects" method, unless you know what you're doing. Depending on your site configuration you can either use "Write directly to files" or "Use FTP", supplying your FTP connection information as well. Most users will need the second (FTP) option.
After you're done setting up Kickstart, hit the big green "Start" button. You will see your browser reloading the page many times; this is normal. On the final page, click the second link titled "here". As the page reads, you're (almost) done! If you had an .htaccess file on your site, it is now renamed to htaccess.bak. Using an FTP client, rename it back to .htaccess and you're really done, in - virtually - no more than a few seconds!
Try it! I just upgraded two sites in 3 minutes flat (including upload times!).