With that budget you are limited to integrated graphics. I would get the following 15.6" Toshiba Satellite laptop for $449 (excluding Office 365). It has a core i5-5200u and Intel HD 5500 which is better than anything with an AMD CPU in that price range. It has a 1080p touchscreen, but the Intel HD 5500 is not powerful enough to play games at that resolution with the exception of LoL as far as I am aware of. You will be limited to 1366x768 or 1600x900 depending on how graphically intensive the game is. You will mostly be playing at low graphic settings, but medium is possible, but that again depends on the game. LoL can be play at 1080p with very high settings because it is not graphically intensive...