Due to lack of zoom (even close deer look far away), I do not recommend a GoPro for anything other than weapon mounting or close quarters pig stalking. Tried it, hated it.
I have a Panasonic HDC-TM900. I don't think they make them anymore, but I'm really happy with the Panasonic line and I'm sure they have one to fit your budget. Mount to a camera arm strapped to a tree and you're ready to go.
Here's a short clip from the camera. Switch to HD in the YouTube viewer.