Theoretically, the ASRock QC5000-ITX/PH will get you to
4K x 2K (4096x2160) @ 30Hz
But I don't know. I've built seven or eight rigs with that board inside of it and I would say my confidence level is low. The Pentium J3710 - even less so. It doesn't support DP 1.2, so it can't do "real" 4K. AsRock claims
4K x 2K (3840x2160) @ 30Hz
For that board.
All of this is really low-power stuff, as you might expect from an embedded system. A non-embedded equivalent with slightly more oomph, which might just be enough to drive a 4k30 display, would be something like an Athlon 5370 on the ASRock AM1H-ITX motherboard. But that STILL doesn't get you hardware RAID.
HDMI 2.0 appears to be available on Skylake motherboards, or at least some kind of HDMI technology which offers full 4K, but unfortunately it doesn't get you 30hz; only 24hz.
There's no one motherboard with an embedded chip on the market that gets you all of what you want.
BUT, there is a very similar solution:
PCPartPicker part list: http://pcpartpicker.com/list/GGs7TH
Price breakdown by merchant: http://pcpartpicker.com/list/GGs7TH/by_merchant/
CPU: AMD A8-7600 3.1GHz Quad-Core Processor ($71.99 @ SuperBiiz)
Motherboard: MSI A68HI AC Mini ITX FM2+ Motherboard ($81.98 @ Newegg)
Memory: Klevv Urbane 8GB (2 x 4GB) DDR3-2400 Memory ($35.99 @ Newegg)
Other: Belkin Miracast Video Adapter (Supports HDMI 2.0) ($59.99)
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-08-02 14:55 EDT-0400
This fits in your current case and runs on your current PSU, with your current storage solutions. It supports RAID 0/1/10 off the FCH's SATA controller for all four possible SATA connections, and, while it cannot do a hardwired 4K connection, it supports a CPU and wireless solution with the oomph and the technology necessary to drive a 4k60 connection to your TV in true, full 4k resolution, with the help of the Belkin Miracast Video Adapter. The wireless card connects in the 5ghz frequency range with the adapter, which is plugged directly into the HDMI 2.0 port on your TV, and then you enable miracast using a charm in Windows 8.1 or 10 (or on any Android device with an OS of 4.2 or higher). Apparently this also works in linux via the MiracleCast software, but I'm not confident in that. It is true that this would require a second connection (either using the built-in ethernet or another wifi adapter in a USB or PCI-E port) to actually get online content onto your screen, but I'm guessing due to the RAID requirement that this is not a big deal.
Of course, this is NOT embedded and NOT passively cooled, so if that's a dealbreaker then I guess we are SOL. I'm hoping that it isn't since your current build seems not to be.
This was the only solution I could find which comes close to what you're looking for. It is true that you could use this dongle with an Intel system that supported Miracast, and there may be other similar HDMI 2.0 miracast adapters out there, but this was the configuration which seemed best-supported and most cost-effective from my point of view. Let me know if you have any questions.