When we start up the computer It boots up to the point where the desktop appears briefly. Then the screen goes blank/dark and nothing else occurs. If shut down it will strat up and get to the desktop display and shut down again.

back light in the screen is bad...
replace the screen

try starting in safe mode.

when you power on the PC, start tapping the F8 key on your keyboard.

this will give you the option to start in safe mode and can possibly let you find out why you cannot start up.

if it does start this way, you should at least be able to retrieve the files you were looking for.

this problem might be due to the corruption of logon file
install a fresh copy of windows or try to restart it in safemode by pressing F8 repeatedly before boot screen.

If you mean after the computer performs the self test the screen goes blank and stays blank, eventually turning off you are having a hard drive problem. When the computer boots you should see an E-Machines logo and/or a self test on the screen. That is done by the motherboard as part of every boot up. After that the motherboard passes control to your boot device and loads your operating system. If your computer stops booting at that point, your boot device is not working properly. That could be caused by many things - virus, hardware failure, power outage. If you really mean you see nothing at all on the screen when you press the power button your computer simply is not working at all. Open the case, blow all the dust out of it and reseat all components you can, then try again.

Try your old HDD in it and see if that works. If it does work then nothing is damaged except your DVD drive/Controller. Try putting the new HDD in another computer, installing windows then putting it back in the HP. If all else fails then flash the bios with an updated image and see if that rectifies the problem.

the easiest solution would be to re-install windows and hope it is not a hardware problem

If tried the monitor on two other computers with the same result -- definitly your monitor is faulty. Take it to the repair shop. To make double sure, run your computer with another monitor. If everything is fine now; you can blame the monitor.

one of the files in the system32 directory in the windows folder is corrupt, the files in the system32 folder contains vital files for the boot procedure, try to start the machine in "safe mode" by pressing F8 after the sony boot screen, if it allows you into the operating system you will need to choose "system Restore" take the time back to at least a week before your problems started DO NOT CREATE A RESTORE POINT

This sounds like a burned out backlight lamp.  Had that happen to a Mac laptop used at work:  all the boot-up LEDs and sounds occurred, but a dark screen.  It was fixed promptly by our computer guy, who is pretty amazing at fixing computers.  So all I can tell you is that it shouldn't be a particularly expensive repair.

THROW IT AWAY AND BUY A NEW ONE