I need suggestions for the best mounting method of a GoPro camera on a Hanger 9 Alpha Electric Aircraft (high wing trainer style with tri-cycle landing gear). Thanks for your suggestions.

I don't know about the best way. Probably depends on what view you are going for. Some people mount the bare camera directly to a smooth surface via a good quality velcro. I didn't want to risk a scratched lens,or worse,so I decided to keep the camera in it's protective housing. For this,hopefully you kept the black plastic display platform(big flat square with the gopro mount in the middle). With that,you can mount it on top of the center of the wing,with some foam rubber underneath to keep from marring the plane,and run extra rubber bands around the platform from various directions until it feels real secure. Cross some around the camera,so there's no way it can slide out from underneath them. You'll just have to experiment a little to find what feels good. It has been awhile,but I did this once with my Avistar trainer. I do remember it handled different with the big "air brake" on top. Struggled more to get to altitude,and may have required up trim for level flight. Luckily I had plenty of power with a .46 engine on it.
I also used this method to mount it on the nose of my Pulse XT 40. Just plenty of rubber bands,and I may have used velcro straps on the Pulse for added security.
http://vimeo.com/15595629

Thanks for the comments and ideas. The video looks great - does the GoPro have the standard lens (sort of "fisheye" 170 degree). I will probably mount down facing for aerial shots and forward for other views.

Yes,standard lens. I was filming on the 720 at 60p setting for that video. It also has a 1080 setting which gives you a noticeably better image with less fish-eye,but only 30 frames per second(which shouldn't matter if the intended use is streaming media). That setting is definitely worth a try before going to the trouble of swapping out the lens(I assumed you had an HD version of the GoPro like mine).
For down facing,I think an under wing mount would be the way to go. But I do not have any good ideas on doing that. It might be a bit bulky with the housing and gopro mounts.