How can I read the client's machine/computer name from the browser?

You can do it with IE 'sometimes' as I have done this for an internal application on an intranet which is IE only. Try the following:

function GetComputerName() {
    try {
        var network = new ActiveXObject('WScript.Network');
        // Show a pop up if it works
        alert(network.computerName);
    }
    catch (e) { }
}

It may or may not require some specific security setting setup in IE as well to allow the browser to access the ActiveX object.

Here is a link to some more info on WScript: More Information

Browser, Operating System, Screen Colors, Screen Resolution, Flash version, and Java Support should all be detectable from JavaScript (and maybe a few more). However, computer name is not possible.

EDIT: Not possible across all browser at least.

Well you could get the ip address using asp.net, then do a reverse DNS lookup on the ip to get the hostname.

From the ASP.NET Developer's cookbook ... Performing a Reverse-DNS Lookup.