A general statement:
Unless you are doing some thing like editing videos this machine is certainly not past it if you replace HDD with a SSD and increase RAM to the 8GB max.
Now to troubleshoot:
Service manual - http://h10032.www1.hp.com/ctg/Manual/c02664684
* Check the video cable from the main body to the hinged screen. Sometime the video will work if the screen is only partially open. Try this in different positions.
* Remove battery and unplug and press power button to discharge any power left in system.
* Open the bottom and blow out all dust. Dust can cause over heating.
* Reseat the RAM modules.
* Take out the CMOS “coin” battery.
* Reassemble to see if screen comes on
That’s a start.