I built a picavet ("peek-a-vet"), named after its French inventor, Pierre Picavet. Properly done, it results in a stable platform on which to mount a camera, and the platform remains horizontally stable (more or less) while in flight, even if the rest of the kite is moving. There's plenty of instructions online for building one, but what they have in common is a cross shape, and at end each of the 4 points of the cross, you have some sort of eye for the routing of line. The routing of the line is important, as this is the mechanism by which the platform is kept stable.
Below is a picture of mine. Last year, with the rush of preparing, this was one of the last projects before the final push to leave...I think I made this just a few weeks before I left. I had some spare aluminum sheet, eyes, fasteners, and bought the small blocks because, you know, it's a sailing picavet. I also didn't want corrosion or salt from creating friction and impeding the movement of line while its working, and thought the blocks would help minimize that. I drilled a small hole at the top and mounted my cheap Go-Pro knock-off to it using just the accessories that came with the camera (an "SJ Cam"). To clip onto the kite's line, I used regular hardware store "S" carabiners, but I tied an alpine butterfly loop onto the kite line and cliped the picavet into that to avoid losing my camera if the rig went down.
As far as kites go, I bought a 165cm x 104cm sled kite, which I had read is the most stable kite for this application. However, sometime on the outgoing leg, I must have damaged it, so couldn't use it on the outbound leg. While in Kaua'i, I went to the only store that seems to carry kites (a toy store in the Hanalei Bay area) and bought the one you see in the video, which definitely was not as stable as the sled kite I intended to use....the one in the video was the largest kite they had in stock. I removed the thin line it came with and instead used parachute cord, which I had 500 feet of onboard.
The whole rig, picavet, camera, and parachute cord was a bit heavy for this kite, but I launched it in ~15 knots and it seemed to do ok.