1) I get the video signal from the composite output of my cable box (or any video source), which outputs HDMI and composite in parallel. If that weren't the case, I could have used an HDMI splitter and HDMI2Composite converter. I prefer the composite signal, as its not encrypted, and you don't really need to waste processing on an HD resolution signal. The "resolution" of your LEDs around your display doesn't even come close to even an SD video signal.
2) The composite cables run into a USB Video Capture card, which I picked up for pretty cheap on Amazon. The USB card is plugged into a Raspberry Pi, running Raspbian.
3) I got the driver for the video card working on Pi. Then wrote a driver for the LED strip, which communicates over SPI.
4) My main program, which is set to run on powerup, will make the video card sample the video signal as fast as it can. I do some image processing on the capture to average the pixel colors in each of several rectangular areas around the border of the image, each assigned to a corresponding LED. Then I send the SPI signal to drive the LED to that color. The sampling, image processing, and LED driving is plenty fast that the LED "frame rate" is well above perceptible limits.
5) This is all configurable with a config file that accepts parameters for LED layout, how big the area to process for each LED should be, overlapping those areas for smoother color transitions, how many frames to average the colors over (also for smoother transitions), etc.
6) The wiring is the simplest part. Power source, split off to a usb connector to power the Pi, the other line split into power and ground for the LED strip. The strip needs 2 lines for communication as part of the SPI protocol, which are just wired to appropriate GPIO pins on the Pi.
Boom. Done.