I have a business class 8300 Elite Desk SFF that I am using for a media room. What would be the cheapest solution to obtain 4k output with a low profile solution? I'll be using Display Port if possible.
Basically the cheapest card that will support 4096 x 2160 will do. Playing videos requires very little GPU processing power and memory. My GPU basically sits at idle when playing videos full screen (in 1080p). If it's PCI-E with at least 1GB RAM and supports 4096 x 2160, it will play what you need.
This might be the better bet depending on how big of a deal the onboard DisplayPort is to you. Looked around quickly and there are a few GT 720s/730s that can output 4K, like the MSI card Lochekey posted, but haven't seen any with DP.
A 750 will definitely do it, but if you want something more than basic upscaling (you will!), a somewhat more powerful GPU is nice to have. Something like a 1060 might be cheap enough and be plenty powerful or you can look for a cheap 960 or wait for the 1050. It all depends on how much of a videophile you are.
AMD is a poor choice for high end video playback. They at least support VDPAU nowadays, but I don't think they have implemented the high quality scalers. The real dealbreaker is that the driver quality is still not that great.
The Shield TV is hard to beat especially if you can get it on sale, but it's not going to match a good PC for upscaling quality. Of course, a good PC also costs a lot more.