I have a Walmart schwinn trailer that I really like...it was something like $150. It is reasonably light (aluminum frame) and has 20" wheels. It has a "trunk" area where you could store some of your stuff and there's plenty of room for a dog inside. You might want to make sure the floor is comfortable for the dog to lay across - on top of your sleeping bag and pillow maybe?

If I was dragging a 65 lb dog up a hill, I would look for looooow gearing - like 22 or 24 small ring on a triple and 30+ tooth big cog in the back. Any bike that can hold a couple of racks, fits you and is comfortable will work. It doesn't have to be expensive - just comfortable and properly prepped. Make sure the hubs and crank are adequately greased. I would look for a 90's mountain bike and put slick tires on it...and maybe some bar ends to give yourself a couple different hand positions.