You just need to upload the files in the WP installation ZIP file. After it’s been un-zipped, of course (I’ve had someone complain it didn’t work when they uploaded the ZIP file to their hosting account). Apart from that the only files that you’ll need to upload are any extra themes or plugins that you want to add from the start. Don’t worry about a .htaccess file as WordPress will generate that when it’s needed.
As long as youre wp-config.php file has got the correct database connection details you will be able to install the system. I wouldn’t be to concerned about the lack of a ‘public_html’ folder. There’s a good chance that your client doesn’t use one of the more popular contorl panels, or possibly not evne any control panel, and that’s jsut how their server is configured. If you’re worried that it’s not the right spot, put up a dummy index.html file and check that you can view that through their websites URL before you do anything else.