Assuming that both you and your customer have Cpanel style hosting, this is what you do:
1. Go into the file manager, and select the directory that your blog is in, and add it to an archive file.
2. Download the archive file.
3. Go to phpMyAdmin, and select the database that your blog is using.
4. Click on the Export tab.
5. Select "Save as file".
6. Click "Go".
7. Open the archive file, and edit the configuration file, so that your server database user and password information is zeroed out (otherwise you might have handed your customer the keys to your site)
8. Re-archive the files as needed (make sure the archive contains the updated configuration file).
9. Send files to your customer.
Uploading:
1. Create database.
2. Execute SQL statement (upload the SQL file, or copy and paste its contents into the SQL statement window).
3. Upload archive (using file manager).
4. Again using file manager, extract archive.
5. Edit configuration file to have correct username and password for the newly created database, plus new domain info and email info, etc. as needed.
6. If no mistakes were made, the new blog should now be virtually identical clone of the original.