I would do it like this: if you're using cpanel or a panel that allows you to compress, then go to the current sites root folder, select all files and folders and click the 'compress' function. If you don't have a compress function, then go here: http://www.net2ftp.com/ and enter your ftp details.
Log in via net2ftp and select all files and folders, then select compress.
Now download that comrpressed folder to your PC. This is a back up of your sites current files, and won't affect the current site which will still be online.
Now create a folder for wordpress in the root folder and install wordpress. So now you should have your current site and wordpress at eg. www.site.com/wordpress.
Edit your site until you're ready for the move, then delete all the files and folders in the root directory EXCEPT the wordpress folder. Open the wp folder, select all files and folders, then select move, move them to the root directory and check the site in your browser. As i really don't like wordpress, i've never installed it, so not sure if there is anything about the config that relies on the precise url, but if the site doesn't work properly, wordpress creates a config file during installation. Check this to see if anything refers to the exact url and change it if necessary to your new root. Also, before moving it, check the site settings and if there is a url setting, change it to the new url before moving.
Apart from the config and site settings there should be no problems in moving it, but feel free to add me and email me if you do get problems.
As a suggestion, if you can add sub domains to your current account, why not create a sub domain, create a folder for wordpress inside the subs root then do the install and move the files as a test run? If it all works, you're ready to go for it on the main domain!