To each their own is my philosophy. Sometimes if you are not a tech type person it is easier to buy a premade system, sometimes it is cheaper and easier to do so even if you are a tech type person.
That said, I would recommend starting with the purpose of the machine. If it is a general game there are lots of budget type items. If it is gaming, I would suggest going with a good CPU, GPU, and RAM over all other items. I went with a i5-3470 (about $190 with shipping/etc), a B75 chipset (B75ma-g43 MSI motherboard) for about $50, 12 gb RAM for about $80-85, a 700w Power Supply for about $35, a ATI/AMD Radeon HD 6570 for about $40, and then reused a DVD-RW (about $10 to buy another one I was going to get), a 500gb WD 7200 RPM hard drive for $25, a standard mouse and keyboard for about $15-20, a 23 in HD TV using HDMI to connect to the computer for about $100 including the HDMI cable, Win 7 Ultimate (varies, I got mine for $70), a all in 1 card reader (43-in-1 with 1 usb 2.0) for about $4, a wireless usb dongle for 150n series connection (about $5).
The CPU, GPU, wireless adapter, TV, card reader, power supply, mouse/keyboard, and RAM were all brand new, motherboard was refurbished, and other parts were mostly lightly used...
I am not a hard core gamer, mostly just office type applications, watching videos, editing pictures, etc for normal home stuff but I do play Mechwarrior Online with it set to high I can still play but I put it at medium and think it runs better and very well.
--EDIT--
I believe that puts the total around $500-550 which is below your price mark, a nicer video card is still less than $100 and capable of decent gaming.