Your code is not working because of an error in this line:
ComputerName = obj.shell.RegRead(regComputerName)
Instead of obj.shell you should be referencing objShell. It should look like this:
Set objShell = WScript.CreateObject("WScript.Shell")
strRegKey = "HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\ComputerName\ComputerName\Computername"
strComputerName = objShell.RegRead(strRegKey)
WScript.Echo strComputerName
However, there are much more reliable ways of getting the computer name without having to deal with the registry.
From WSH (as suggested above)
Set WshNetwork = WScript.CreateObject("WScript.Network")
strComputerName = WshNetwork.ComputerName
WScript.Echo "Computer Name: " & strComputerName
From an environmental variable...
Set wshShell = WScript.CreateObject("WScript.Shell")
strComputerName = wshShell.ExpandEnvironmentStrings("%COMPUTERNAME%")
WScript.Echo "Computer Name: " & strComputerName
From WMI...
strcomputer = "."
Set objWMISvc = GetObject("winmgmts:\\" & strComputer & "\root\cimv2")
Set colItems = objWMISvc.ExecQuery("Select * from Win32_ComputerSystem",, 48)
For Each objItem in colItems
strComputerName = objItem.Name
WScript.Echo "Computer Name: " & strComputerName
Next
From ADSI...
Set objSysInfo = CreateObject("WinNTSystemInfo")
strComputerName = objSysInfo.ComputerName
WScript.Echo "Computer Name: " & strComputerName
From ADSI (only works for domain members)...
Set objSysInfo = CreateObject("ADSystemInfo")
strComputerName = objSysInfo.ComputerName
WScript.Echo "Computer Name: " & strComputerName
...and one last way for Windows XP users only...
Set objPC = CreateObject("Shell.LocalMachine")
strComputerName = objPC.MachineName
WScript.Echo "Computer Name: " & strComputerName