Are you buying this just for league? A desktop is the way to go.
Tell us what are your expectations with gaming? What resolution are you going to run? Do you play the newest, most intensive games at high resolution? If this computer is mostly for league and you aren't playing the most recent, high intensive games, then I would go with:
4GB ram (no more needed unless really really hardcore gaming or high cpu intensive programs like video editing...if you are using these types of programs, you would want 16GB)
entry level ati video card 100-150$. Entry level ati/amd graphics cards are a lot better bang for your buck than nvidia. If you go higher than 150$, go with an nvidia card.
entry level cpu go with intel. i7 is overpriced. The price/performance is a big amount of decreasing marginal returns with i7 processors, which means you pay a ton more for not that much more increase in performance. Best bang for the buck is the cheapest i5 or a mid level i3 processor. Those are way more than enough for league.
motherboard: get one with solid state high quality capacitors. entry level asus are good boards. motherboard socket must match the processor and ram type. so 1st decision you must make with a new build is the processor. Then, find the motherboard, ram most appropriate for that cpu.
hard drive: ssd preferred, but can sata
case: enough room for future expansion
power supply: 500w
monitor/mouse/keyboard: relative to what resolution you use, and how much you game and prefer a certain mouse/keyboard.
These specs would be good for what you are looking to accomplish if you want entry level and the computer is mostly to be used for league. It would also be able to handle other games, and then you could upgrade the graphics card in a few years if you are playing more intensive games. It is better to buy an entry level video card for 100$ and then buy another entry level video card in 3 years for another 100$ than spending 350$ originally when building the pc. :)