My choice would be an AMD Llano CPU for this usage. The integrated graphics are fantastic for HD content and still good for casual gaming. Not having to add a graphics card will also keep the power consumption, noise and heat to a minimum which is always good in a HTPC.
This is at the top end of your budget but should be fantastic for your usage:
AMD A8-3850 OEM
G.SKILL Value Series 8GB (2 x 4GB)
ASRock A55M-HVS FM1
Crucial M4 64GB
Seagate Barracuda 500GB
LIAN LI PC-C37B
SeaSonic SS-300ET 300W
I personally consider the SSD and case to be 2 of the more important parts in this kind of build but if you don't see it that way, you could always get a cheaper case and drop the SSD. You really only need 4GB of RAM too.
If you drop the SSD, go down to 4GB RAM and change the case to something like the ML03B, the price will go down to around $500. It would just mean the system would be slightly noisier and slightly less responsive and have a plastic front panel instead of aluminium.