First of all I should tell you that just about any potato can run LoL maxed out at 60fps at 1080p/1440p/1600p. That being said...
CPU - Stay away from AMD APUs, they're meant for low end system that just need the basics. Stay away from AMD CPUs in general.
HSF - Meh, whatever. Hell you could even use the stock one since it doesn't sound like you want to OC it... do AMD APUs even come with one?
Thermal paste - MX-2 is fine.
Mobo - Too many options here based on what CPU you pick, just don't stinge out then regret it.
Memory - Don't forget to check timings. DDR3-1600 is the standard, but for about the same price you can get DDR3-2000 or 2133.
GPU - This is where it's impossible to give a recommendation, any piece of crap can max out LoL but buying such a video card like an R7 250 seems like a waste of money it's so weak.
PSU - This is the part where everyone stinges out which happens to be the LAST part which you should stinge out on. I don't know what the draw of something like the R7 250, but regardless I always recommend overkill. The closer a PSU is to its rating the less power efficient it is. For example my 950W PSU would have to draw probably around 1.3KW in order to supply 950W, however to supply say 600W it wouldn't need to draw any more than 750W.
Wireless - Whatever, though I would be weary of Rosewill stuff (Newegg's house brand).
If you're on that tight of a budget and you insist on an AMD APU then get whatever their most powerful one is if the GPU on it is powerful enough to max LoL (it should be) and skip the discrete graphics card altogether.
