For gaming, even more so for live streaming, I would not recommend a Pentium. I'd be looking towards the i5 territory. If you're willing to purchase used parts, for less than £400 I can build you the best desktop "bang for your buck." Only the processor and GPU are used, everything else is brand new. (To be honest, the CPU is even in new condition.)
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Core i5-4570S 2.9GHz Quad-Core Processor (£113.20 @ eBay)
Motherboard: Gigabyte GA-B85M-DS3H Micro ATX LGA1150 Motherboard (£44.98 @ CCL Computers)
Memory: Kingston 8GB (2 x 4GB) DDR3-1600 Memory (£56.99 @ Ebuyer)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive (£35.94 @ Aria PC)
Video Card: MSI GeForce GTX 560 Ti 1GB Video Card (£56.00 @ eBay)
Case: Cooler Master N200 MicroATX Mid Tower Case (£30.36 @ CCL Computers)
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply (£35.38 @ Amazon UK)
Optical Drive: Samsung SH-224DB/BEBE DVD/CD Writer (£10.09 @ CCL Computers)
Total: £382.94
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2014-10-13 00:33 BST+0100
The GTX 560 Ti performs similar to the R7 260X to be honest anyhow; they both have DX11 and will have partial DX12 support. The above build is much better in performance over your otherwise brand-new Pentium build. The case I've selected isn't the most attractive, but it's cheap, includes two case fans and has USB 3.0 supports (so does the motherboard).
By the way I have also included an optical disc drive. You can remove it if you believe it's unnecessary; thus saving additional funds.
All the best.