I am not an expert but it sounds like its to do with your computer. First check here, https://www.wizard101.com/game/faq# for the requirements to run the game on your computer. If all of that checks out, I'd definitely find out what is wrong with your computer. It could be something as simple as your firewall.
I get on my character and right when I get on my computer will shut off. What the heck is happening?
How should I know? You never said what kind of computer you have, what it has for graphics, and memory. You said you computer shuts off, now is that like reboot or does it actualy shut off completely?
But by the little info you gave me IMO; I would have to say the reason why your computer shuts off is that your CPU is over heating causing the emergency circut inside it to shut off the power so it won't melt and cause a fire.