If you plan on building one yourself, here is what I got;
CPU/GPU; AMD A10-6800K 4.1GHz Quad-Core with Radeon HD 8670D
149.99 on newegg
Motherboard; GIGABYTE GA-F2A85XM-DH3
Can not find a price for this for the life of me. I can assume its around 80 dollars.
Storage; Western Digital Caviar Blue 1TB 3.5" 7200RPM
52.99 on newegg
Antec VSK3000E
$31.11 on newegg
Power Supply ; Corsair 500W ATX12V
$54.99 on newegg
RAM; G-skill F3-2133C10D-8GXM
$82.99 on newegg
OS; windows 8.1 OEM
$99.99 on newegg
Notes;
My P.C is similar to this one, but not the same. This build is hypothetical
However I do use the same cpu, RAM and motherboard, and used the same case so it should work.
First things first;
The CPU runs league almost maxed out 60 FPS. If you decide to buy an after market cooler (probably a liquid cooler, as the motherboard does not offer much room for a large air cooler in there, for example corsiar h100) It's a very expandable overclock able CPU.
want a dedicated gpu? Knock yourself out. You don't need one for league though. A 7770 will work. I would not get one, I got it before the amd R series cards came about, the 7770 is outdated now. Remember that the motherboard has support for two cards, in case you would need.
Storage is pretty personal, 1 TB is kinda overkill, and 7200 RPM is fairly high for a HDD. You can also get an SSD if that floats your boat, a little expensive though :'(.
Case worked. It looked like shit, sounded like shit, smelt like shit, preformed like shit, it was shit. You are on a budget, so i would imagien you don't care about the case. Just pop your parts in and hide it under the table, and it works fine. I use a corsair m240 air. It works wonders, but costs 100 dollars :/
PSU is also fairly overkill, but you will probably want to get some nice GPU's in the future.
RAM. You 8 gigs would be smart if you are not getting a dedicated GPU. Considering how much windows will take up, then your integrated graphics, then your game, then your porn/groove-shark it would be wise to spend the money on 8.
Adds up to around 550 dollars (including OS) This is on newegg. If you buy your parts in person, get ready to pay more.
I hope this helps.
