I'd definitely like to start with our AMD gaming PC build.
Best Budget AMD Gaming PC Build Under $800 [October 2012] | Custom PC Review
To shave off $100 right off the bat, we can go with the HD 7770, which is currently running for $100 after rebate. It'll play Minecraft just fine and a bit more as well.
Amazon.com: Sapphire Radeon HD 7770 GHZ 1 GB DDR5 HDMI/DVI-I/DP PCI-Express Graphics Card 11201-17-20G: Computers & Accessories
For the motherboard, we can drop down to the GA-970A-DS3 motherboard.
Amazon.com: Gigabyte GA-970A-DS3 AM3+ AMD 970 SATA 6Gb/s USB 3.0 ATX AMD Motherboard: Computers & Accessories
We can also drop the power supply to the Antec VP-450 which will work fine.
Amazon.com: Antec VP-450 450 Watt Energy Star Certified Power Supply: Electronics
With these changes, you should be able to get the system slightly above $600. I wouldn't recommend dropping specs down much farther than that since you're looking to stream and record.
For the monitor you can just got with this ASUS. It'll come with speakers, which will suck but it should hold you over until your budget opens up for better speakers.
Amazon.com: Asus VE228H 21.5-Inch Full-HD LED Monitor with Integrated Speakers: Computers & Accessories
Otherwise, a mouse and keyboard I say just go for something cheap until you can afford something decent. I can make some recommendations, but it'll depend on how much budget you have left.