What is the best/highest quality gaming PC I could build for $300?

You can't really build one for $300. BUT, I can tell you I built mine for around $350.

AMD APU = $90
Motherboard = $50/60
Memory = $40
PSU = $30
Case = $30
OS = $90
HDD = $30

Total = $370
*Edit added hard drive

This will run most games at low settings depending on the parts you pick. Now a lot of people will not recommend building this computer as it won't be very powerful. But, if your really tight on a budget sub $400 systems are possible - just need to decide for yourself if your O.K. with low settings and the possibility of the system not being useful at all in 2-3 years.

If you decide to actually try to build for around $350 remember to wait for sales/combo deals. I started contructing mine in early November and finished in mid December because I wanted to be as patient as I could with the deals.
 
Under $300 is impossible, but this is only 311. It won't really play very much either way though.

PCPartPicker part list / Price breakdown by merchant

CPU: AMD A6-6400K 3.9GHz Dual-Core Processor ($47.99 @ NCIX US)
Motherboard: Gigabyte GA-F2A68HM-HD2 Micro ATX FM2+ Motherboard ($43.98 @ OutletPC)
Memory: Apotop 4GB (1 x 4GB) DDR3-1600 Memory ($27.11 @ Newegg)
Storage: Western Digital Caviar Blue 320GB 3.5" 7200RPM Internal Hard Drive ($28.10 @ Amazon)
Case: Cougar MG100 MicroATX Mini Tower Case ($28.99 @ SuperBiiz)
Power Supply: EVGA 430W 80+ Certified ATX Power Supply ($32.99 @ SuperBiiz)
Optical Drive: Lite-On iHAS124-14 DVD/CD Writer ($14.09 @ Newegg)
Operating System: Microsoft Windows 8.1 (OEM) (64-bit) ($87.79 @ OutletPC)
Total: $311.04
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-04-24 13:08 EDT-0400
 
Get 530 and you have a system that will actually play games, example> https://pcpartpicker.com/b/wxccCJ

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i3-4150 3.5GHz Dual-Core Processor ($103.99 @ SuperBiiz)
Motherboard: MSI H81M-P33 Micro ATX LGA1150 Motherboard ($43.99 @ SuperBiiz)
Memory: Apotop 4GB (1 x 4GB) DDR3-1600 Memory ($27.11 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($52.49 @ OutletPC)
Video Card: Zotac GeForce GTX 750 Ti 2GB Video Card ($129.99 @ SuperBiiz)
Case: Cougar Spike MicroATX Mini Tower Case ($29.99 @ Mwave)
Power Supply: EVGA 430W 80+ Certified ATX Power Supply ($32.99 @ SuperBiiz)
Optical Drive: Lite-On iHAS124-14 DVD/CD Writer ($14.09 @ Newegg)
Operating System: Microsoft Windows 8.1 (OEM) (64-bit) ($87.79 @ OutletPC)
Total: $522.43
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-04-24 17:48 EDT-0400
 
Also could I use the AMD Phenom II B59 3.4GHz Dual-Core AM3 Processor (mainly cause I'm trying to cut down a little on price wise) would it work just as well as the Intel processor?
 


No it wouldn't be anywhere near an intel processor. Cores and GHz don't even tell 50% of the story of how a CPU will perform. Gaming PCs start at about $500, anything below that is just gonna be a PC for browsing the web.
 

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