Where you do your CS degree doesnt matter really. People get into games from all blends of degree. Important thing is just to learn programming really well (dont even touch making a game until your comfortable with C++). Its alot easier if you do it this way . IMHO I think people often start to try making games (im gonna make gears of war!!) and they cant write simple C++ programs, therefore their motivation is lost when they hit the first wall.
Then start making demos. My advice would be do it from scratch. Dont use 3rd party engines etc, because you’ll learn very little. i.e. a demo built using a pre-built engine wont get you a job (unless your demonstrating ai techniques or gameplay systems or networking you wrote from scratch and the graphics engine only facilitates showing off your coding skills). Similarly, if you want a graphics job you need to learn the hard stuff from scratch, a demo made in unreal\source unfortunately wont do it. Bizarre creations have great advice in this respect on their website.