Assumptions

• You've created your database already and know that database information.
• You're installing your 'main registered' domain (not an add-on domain) and that you want visitors to access your WordPress blog at www.sample.com (use your real domain in place of sample.com).
• You've downloaded the wordpress-x.zip (where x is the version) file to your local machine.

Using the cPanel Legacy File Manager

• Navigate to your web-root folder (public_html)
• Click on Upload Files link, then upload the whole wordpress-2.3.1.zip into your web-root folder (public_html).
• After uploading, click on that file name, then click on Extract File Contents, and the contents will extract to a folder called wordpress.
• Click into the wordpress folder, rename wp-config-sample.com to wp-config.com.
• Click on the wp-config.php file then Edit file. Put your database information in that file. DB_HOST is probably 'localhost'.

Using your browser (like FireFox)

• Browse to www.sample.com/wordpress. The installation process will start
• Enter the information to complete the install.
• Login with the user/password provided
• Go to Options->General and change the Blog address (URL) to http://www.sample.com (delete the /wordpress at the end).

Now back in cPanel Legacy File Manager

• Make sure you are in the public_html folder (wordpress folder should be there)
• Click on Create New File link, call the file index.php
• Put this information in the file and save it.

<?php
/* Short and sweet */
define('WP_USE_THEMES', true);
require('./wordpress/wp-blog-header.php');
?>

That's it