The following laptop should be fine at playing LoL. It is a 15.6" HP 350 G1. It has the i5-4210 (1.7GHz / 2.7GHz) with the integrated Intel HD 4400, 4GB of RAM (recommend you add another 4GB when you have money), 1366x768 resolution and it comes with Windows 7 Pro, but it also comes with a Windows 8.1 Pro license. It costs $480.
The following laptop should be fine at playing LoL. It is a 15.6" HP 350 G1. It has the i5-4210 (1.7GHz / 2.7GHz) with the integrated Intel HD 4400, 4GB of RAM (recommend you add another 4GB when you have money), 1366x768 resolution and it comes with Windows 7 Pro, but it also comes with a Windows 8.1 Pro license. It costs $480.