I would guess that the spotlight and it's wiring could be left as is and the camera mounted above the light. The wiring for the GoPro camera will have to include power for the camera as well as a data cable to view the images. The 12 volt power can be run to the camera with a mini USB connection and another micro HDMI cable would have to be run to transfer the video to your monitor.
The biggest problem I see is that the Go Pro is not waterproof. They do sell water proof cases for the Go Pro but these are designed to be used underwater and do not have provisions for the two external cables that yo would need.
I love the GoPro camera and use on on my dash but I would think that a home surveillance camera would be better for what you need. Most of these are run on 12 volts and they use a single cable to supply power and video transmission to a monitor. The external surveillance cameras can be found that have 720 or 1080 resolution and are already designed to keep out the elements. Attached to your light they should be able to view the surrounding area as well as the GoPro. Some of the surveillance cameras can be hooked up to their own joy stick and can be rotated, tilted, and zoomed(like they do in the Vegas casinos) to provide even more coverage than you can get from a GoPro. It just depends on how much you want to spend and what resolution you want.