The problem is that with a <300 budget computer it really isn't worth going into the intel route. AMD actually makes some decent cpus which come with gpus, and tbh it would be a hard fit to try and fit all the parts under 300. I can try list some good parts here:
PCPartPicker part list:
http://pcpartpicker.com/p/ZrzMyc
Price breakdown by merchant: http://pcpartpicker.com/p/ZrzMyc/by_merchant/
CPU: AMD A8-6600K 3.9GHz Quad-Core Processor ($89.99 @ Amazon)
CPU Cooler: Cooler Master Hyper 212 Plus 76.8 CFM Sleeve Bearing CPU Cooler ($19.99 @ Newegg)
Motherboard: Gigabyte GA-F2A55M-S1 Micro ATX FM2+ Motherboard ($34.69 @ Newegg)
Memory: G.Skill Ripjaws X Series 4GB (2 x 2GB) DDR3-1600 Memory ($44.99 @ Newegg)
Storage: Western Digital RE3 1TB 3.5" 7200RPM Internal Hard Drive ($46.99 @ Amazon)
Case: NZXT Source 210 (Black) ATX Mid Tower Case ($29.99 @ Newegg)
Power Supply: Corsair Builder 600W 80+ Bronze Certified ATX Power Supply ($34.99 @ Newegg)
Total: $301.63
Prices include shipping, taxes, and discounts when available
Best I could do