HP G60 Laptop Screen Goes Blank?

I have a three year old HP laptop running Windows 7 (64 bit) that is misbehaving. It would not boot up recently, and I believe the root was heat related issues. My son likes to use it while setting on his bed - fan gets covered up and the laptop overheats. I added some large rubber feet to the bottom to improve clearance for the fan and that has helped, to a point. The laptop will boot and I can log into my son's account. After 5-10 minutes, the screen goes blank (50% grey scale, neither bright or completely dark). At that point, nothing seems to bring the screen back, so it is unusable and a hard shut down is necessary. I have cleaned it with Malware Bytes in the past and now have Microsoft Security Essentials installed, so I think/hope any virus issues have been resolved. My suspicion is that this is heat related, but exactly what is failing I do not know. Two months ago I replaced the power jack in the laptop and the power supply because of power connection problems. The battery is shot and will not hold a charge, so the laptop can not be used unless it is connected to the power supply. Since I replaced the power supply and jack, the laptop has been functioning okay. It would crash on occasion, but I attributed that to malware. I had to purge the hard drive earlier this year and restore the system because of a malware infestation.