If you have prepared yourself to spend $750, I would spend maybe $300-400 on the parts and then spend a few extra on making it silent and cool.
Something like:
Fractal Design Core 1000: $36
Corsair CX430M: $40
Gigabyte F2A85XM-HD3: $65
AMD A6-6400K: $75
Corsair Vengeance 1600MHz 2x2GB: $52
The amount of HDD/SSD space you want: $60-120
Perhaps a CPU cooler for silence: ~$25
Even more silence, Corsair AF120 QE: $18 (to replace the stock intake fan. This one comes with a 12v->7v adapter and is more effecient)
Optical disk drive (if you need it): ~$25
Most expensive option for above: $460
Would play any video out there.
You could go for faster RAM, which you should. But I'm at work, so I can't search through Newegg to find a good deal. Just went with what I know is good. I think G.Skill usually makes cheap 2133MHz RAM, which would be good on an APU platform. It uses desktop RAM as GPU RAM, so speed = performance.
Most of these boards only have 1 HDMI connector. The one I've chosen offers great features-performance/price in my opinion. A DVI connector is the same as an HDMI, just without sound. Just FYI if it helps (You can get HDMI->DVI cables and you lose no video quality, as the video signal is exactly the same)