Sounds like artifacting (the green dots appearing) and then the GPU gets turned off for protection due to a critical malfunctioning, either overheating, power issue or just a defective hardware.
Try running furmark to stress test your video card, the program will report the GPU temps so you can see if overheating is the issue.
What power supply do you have? a low quality PSU could cause the problems you're having by not supplying the power required when on intensive games.