I have my PC built by someone who has good knowledge on the topic and he got me these:
▬ Cooler Master Silencio 352 - Minitowermodel - micro ATX - no feeding ( ATX /PS/2 ) - black - USB/Audio - SD-cardreader
▬ CoolerMaster GM Series G550M - Feeding ( internal ) - ATX12V 2.31 - 80 PLUS Bronze - 100-250 Volt AC - 550 Watt - Active PFC
▬ MSI B150M Mortar - Motherboard - Micro-ATX - B150 - LGA1151 Socket - USB 3.1 - Gigabit Ethernet - HD Audio
▬ MSI Radeon RX 480 Gaming X 4G - Graphic card - Radeon RX 480 - 4GB GDDR5 - PCI Express 3.0 - 2x HDMI, 2x DisplayPort, DVI-D
▬ LG BH16NS55 - Disk station - BDXL Writer - 16x2x12x - Serial ATA - intern - 5.25" - black
▬ Kingston HyperX FURY Black Series - Memory - 16 GB : 2 x 8 GB - DIMM 240-pins - DDR3L - 1600 MHz / PC3-12800 - CL10 - 1.35V
▬ Seagate Barracuda ST2000DM001 - Hard disk - 2 TB - intern - 3.5" - SATA-600 - 7200 tpm -buffer: 64 MB
▬ Samsung 750 EVO MZ-750250 - Solid state drive - coded - 250 GB - intern - 2.5" - SATA 6Gb/s -buffer: 256 MB - 256-bits AES -
▬ Intel Core i5-6500 / 3.2 GHz - 6 MB cache - LGA1151 Socket - box
▬ Be quiet! Pure Rock - Cooler for processor - PMW Fan - 4 heatpipes - Intel/AMD Compatible
This PC was built a little bit longer than 2 years ago, and it costed me about $1590,00. It's way over your budget, but who knows, maybe these parts became cheaper by now. It still runs fast, can handle top graphics of all games at the highest FPS. Heck, I can even stream like that, although I must admit, then it becomes quite hard for my PC to properly stream.
You can also check if you can buy used components that are less important, like the case.