For your processor, i suggest an intel processor, an i5 will do, they are pretty cheap and some of them are really fast and overclockable if you are into that sort of thing. Even some i3's would be okayish, but i suggest i5 just so you can handle other games a little better.
GPU(or graphics card) Go for an Nvidia card, they are a bit more expensive than AMD Radeon cards, but they run cooler and last longer for the same if not better performance. Plus if you have an issue, Nvidia customer support is excellent. Plus, you get Geforce Experience which some folks don't like, but i personally love it, easiest driver updates ever and you get ShadowPlay to record with based on if your card can handle it.
For league you only need about 8GB of ram to start but will want 16gb for other games, and then you can just get another 8gb stick later, it is better to run dual channel, so if you start with 8gb, get two 4gb sticks until you can upgrade to two 8gb sticks for 16gb.
Your motherboard doesn't have to be anything fancy, it just has to support your video card slot, your processor, and your ram type (ddr3, or ddr4) Be careful here, because it has to match the slot for your processor, and your ram, and your GPU.
Ibuypower sorts this stuff out for you, so you don't have to worry about getting a wrong motherboard for your ram or GPU or processor or anything, so it's pretty worry free, where as building it on your own, you must know what every piece is because some aren't compatible, and some just flat out don't fit.
I believe that should be enough tips, make sure your power supply has enough juice to power everything, if you plan to upgrade in the future, it is smart to get higher than you need, it will not damage your system to have more, but for the sake of the electric bill get something fairly even. And make sure you have decent cooling and air circulation.
However my final tip, just save up a little more than the budget you have now, a couple hundred dollars goes a LONG way. The difference between a $300 machine and a $600 machine is light and day. Again, wait for that sweet deal and be patient man, because i realize now, ibuypower doesn't have very many deals going on right now, so there's not much for your budget. I would save up a little longer if i were you. Hope this helps man GL!