This build is probably as low as I'd go. It would need you to reuse your Hard Drive and optical drive. You would also need an operating system.
http://pcpartpicker.com/p/7F3dBm
AMD Athlon X4 860K 3.7GHz $66.99
Asus A68HM-K Micro ATX FM2+ Motherboard $32.98
G.Skill Ares Series 8GB (2 x 4GB) DDR3-2400 Memory $29.99
Sapphire Radeon R7 360 2GB NITRO Video Card $88.98
Rosewill Blackbone ATX Mid Tower Case $31.99
Corsair Builder 600W 80+ Bronze Certified ATX Power Supply $36.99
Base Total: $333.94
Promo Discounts: -$8.00
Mail-in Rebates: -$45.00
Shipping: $6.98
Total: $287.92
If you had a full atx or even micro atx case then you could have saved a few bucks there, but those tiny emachine cases are a pain to work with. As far as upgrading what you have in that little e machine you could get the max supported CPU, some RAM, and maybe an nvidia GT 730 with a single slot cooler, but it still wouldn't be great and would be slower than what I'm proposing here.