A few of us were talking about this in the HTPC forum recently. As I understand the Fire TV doesn't have an MPEG2 decoder, so it can't natively play cable tv.
XBMC allegedly has a software decoder that would work to display the content. I'd like to test it out myself.
DLNA is probably out due to the MPEG2 issue. The native DLNA client will recognize the stream as something it can't play.
To get XBMC up and running you need ServerWMC running on a PC. That will act as the recorder storage, scheduling client, and live tv buffer.
For it to work at all you need a permissive cable provider like Comcast or Verizon. The channels need to come through Copy Freely or you will never be able to open them.