What kind of steel is it in that axe? If it's mild steel (which I suspect, since it's soft and the paint on it indicates that it hasn't been untempered) heat treating it won't do much, though it'd be better than nothing. A high carbon steel would be ideal, like the kind you'd find in files, grass cutting blades, saws, and stuff like that.
There's some matter of opinion on how hard you want the blade to be, the hardest blades require a diamond grit sharpener, though only once in a while, where the softer blades will need a more frequent sharpening, but you will be able to use found natural stones to do that. I've made a few knives, my first I actually made with only a dremel tool with cutting discs, but that was before I got trained in knifemaking.