I would definitely recommend you go with Intel's i3 or even i5 if you can afford it. The Fusion APUs are only good for extremely low budget builds and HTPC builds.
This is what I recommend for you:
CPU: Intel i5 2400 $190
http://www.newegg.com/Product/Product.aspx?Item=N82E16819115074
This is a great CPU, it's better than the intel i3 2100, which is one of your CPU choices. It has 4 cores and 4 threads. It is perfect for your use because it is relatively cheap, but it cannot be overclocked, which should not be a problem. It has similar performance as the Intel i5 2500k but without the ability to OC the CPU.
Graphics: SAPPHIRE 100314-3L Radeon HD 6870 $165
http://www.newegg.com/Product/Product.aspx?Item=N82E16814102948
This is a great graphics card, it gives great performance for its price. It will allow you to play all modern games. I have this graphics card and it has not let me down yet.
MoBo: ASRock H61M-VS $55
http://www.newegg.com/Product/Product.aspx?Item=N82E16813157241
This motherboard will suit your needs, it is cheap but it also sacrifices some things such as PCIe slots and RAM slots. Overall this is the perfect board for you.
PSU: XFX Core Edition PRO550W $70
http://www.newegg.com/Product/Product.aspx?Item=N82E16817207013
This is a great PSU and has enough power to make your system run. Infact, it should have enough power to SLI/CF, although I don't think you'll be doing that. It is also 80+ certified, therefore, it should be energy efficient.
Case: NZXT Gamma $40
http://www.amazon.com/GAMMA-Classic-Interior-Chassis-Black/dp/B002UDK9U6
This is a great case that is decently priced. You will not find another case with similar features with a $40 price tag. This case was also designed for a gaming computer and therefore it can support longer graphics cards.
HDD: Seagate Barracuda ST500DM002 500GB $85
http://www.newegg.com/Product/Product.aspx?Item=N82E16822148767
This HDD is a cheaper one and is from a reliable company. If you need more storage, then you'll want a larger HDD, but this is perfect for most applications.
RAM: G.SKILL Value Series 8GB $37
http://www.newegg.com/Product/Product.aspx?Item=N82E16820231422
This is a great RAM kit for the money. It is from a very reliable company and is decently priced. This RAM does not have heatspreaders but I doubt you will need them.
Heatsink+Fan: STOCK $0
DVD-ROM: LITE-ON DVD Burner - Bulk 24X $18
http://www.newegg.com/Product/Product.aspx?Item=N82E16827106289
This is a cheap DVD burner with great ratings. This is mainly to install programs that is on discs, but you can also watch DVDs if you want to. If you like Blu-rays better, then you can go with a blu-ray drive.
Total: $660