Black Screen While Gaming

I have had issues with games lately. I recently have bought Starcraft 2 and Rift, the only problem is they black out my screen and lock down my computer forcing a restart. While playing starcraft 2 if I was cinematics or skip them and try to start a match it will give me a black screen with grey lines going down the screen. I contacted blizzard and recieved no word in return. I then tried to start playing rift with some friends. I had played it back at launch and it would randomly crash my computer when I viewed major landscapes. I thought that it may just be new game crashes. I have since tried to play it recently two or three days ago. Everytime I look at an actual Rift my computer does the same thing as Starcraft 2 with the black screen. I have contacted the creators of Rift and they have given me various things to try such as:

GOing and disabling all startup programs off, and disabling all setup programs, excluding microsoft programs.

Next they told me to send them a copy of my DxDiag information.

After they sent me to a link
https://trionworlds.custhelp.com/app/answers/detail/a_id/461/kw/Anti

to change my settings of my graphics card, ATI Radeon 4800 series.

After I changed my settings I was able to look directly at the rift for about 10-15 seconds before my computer screen turned a greyish white and locked up my computer.

Is there anything else that I can do to fix this problem? I believe that I have the required specs for both of these games.

Edit: I have a Dell Dimension E521 that has everything from when I bought it excluding the upgraded Graphics Card, ATI Radeon 4800 Series.