Something like this might work:
PCPartPicker part list / Price breakdown by merchant
| Type | Item | Price |
|---|---|---|
| CPU | Intel Core i3-4130 3.4GHz Dual-Core Processor | $114.99 @ Directron |
| Motherboard | MSI CSM-H87M-G43 Micro ATX LGA1150 Motherboard | $67.49 @ NCIX US |
| Memory | *G.Skill Ripjaws X Series 8GB (1 x 8GB) DDR3-1600 Memory | $49.99 @ Newegg |
| Storage | Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive | $44.99 @ Amazon |
| Video Card | XFX Radeon R9 280 3GB Double Dissipation Video Card | $189.99 @ Micro Center |
| Case | Thermaltake Versa H23 ATX Mid Tower Case | $43.90 @ Amazon |
| Power Supply | Corsair Builder 430W 80+ Bronze Certified ATX Power Supply | $39.99 @ Newegg |
| Prices include shipping, taxes, rebates, and discounts | ||
| Total | $551.34 | |
| *Lowest price parts chosen from parametric criteria | ||
| Generated by PCPartPicker 2015-04-08 03:19 EDT-0400 |
For streaming, an i5 is ideal but it will be tough to fit it in your budget. Assuming you are only playing graphically simple games like League, you could downgrade the graphics card to an R9 270 and upgrade the CPU to an i5-4690k. If you wanted to play more demanding games then you need to keep the 280.