I play a lot of league of legends and Minecraft but my old computer just isn't giving me the performance I am looking for. I am looking for a CPU and mother board that can run both games at playable setting.
The posted one should be good enough, you did mention though that you would only buy processor plus motherboard, did you have ddr3 memory from your old computer or something? if you don't then there is nothing to say that is better that what filippi said, I have a much less powerfull GPU in one of my computers and it can run LoL at 60fps with shadows off, so you should be fine.
If you can stretch the budget for the GPU, I'd reccomend a GT 740 @ $99.99 It should be good enough for most games and low-medium settings.
EDIT: Never mind, Filippi's suggestion for the R7 250x is great, But, if you are planning to play games more Nvidia based, then the GT 740 should be fine, But, games like you stated, Are optimized for both i believe,