I am getting the client\'s (website user\'s) IP address. Now I\'d like to go one step further by knowing the user\'s computer name. So far, my research has not turned up any
If you're referring to the hostname (displayed for instance by the hostname command on linux) of the computer doing the request:
That information is not included in an HTTP request. (That is, it's impossible for PHP to figure out.)
You could do a reverse DNS lookup, but that's probably not what you want anyway.