You will either need to get the Superdrive repaired or replaced in order to run AHT. You can get an external USB powered DVD drive and run AHT from that if you want. When you are able to run AHT run it in extended mode at least 2-3 times back to back. If any errors appear that means it has had a hardware failure. If no errors appear that only means the machine needs more robust testing, something the end user cannot do. You will need to take the machine to an Apple Store or AASP to be professionally diagnosed.
However I suspect your machine may be suffering from a failing graphics card.