What specs should I look for? Laptop versus desktop?
A gaming computer has upgraded components - more RAM, higher speed CPU, solid state drives, and so forth. All this costs a lot of money.
I have been really happy with my new ASUS GL703GE, a 17" notebook for $1199 at COSTCO. The 15" version may be within your price restraint. I also looked at Dell, Acer, Lenovo, and a couple of others. My ASUS seemed to be the best choice at under $1200. Also look at the ASUS GL753GE . . . it has a thicker body and stays very cool.