So if you want to get that old girl cleaned up, which sounds like you have, which is good. Vinegar as stated is an option, along with other means as well. Angle grinder with hard wire wheel is another - quicker, but need a ventalated area - all that rust and metal thrown into the air isn't too good for the lungs. Same if you happen to use a grinder wheel - the old timers in the axe shops that were on sharpening detail unfortunately didn't last long in life - pretty much all died early.
Too sharpen that thing, you will need to secure it down to your work bench, grab some WD40, file of your choosing(I use a Nicholson), and get mentally prepared to be there awhile. Also get ready for a workout for your wrists and forearms if you aren't a regular at this.
Next step then is how to properly file it for a correct axe edge. Just dont go hog wild on it, you need to have the correct angle when finished. Remember you dont want it so it is wedge like, and you dont want it too then at the bit edge. If you have never done it I would tell you to get some learning in on an old beater axe head. Also the WD40 comes in handy to spray not only the axe head clean, but also your file clean of debris.
The angle of your bit for correct sharpening will depend on who you talk to, what book you reference, etc. In my experience, you just know when it is right, and depending on the axe head itself, its makeup, its metal quality, etc, the angle might not always be the same for every single axe head. And more importantly, this is ok. They wont always be the same. You have to look at it closely, and after a couple times you can tell when it is done correctly. You can then put the sharpness to it in a variety of ways. Lots of guys use sandpaper, belts, etc. I just keep using a finer and finer stone until I am happy. The sharpest one I ever got, I got done, and took my thumb and pointer finger and pinched the bit lightly and ran my fingers across it, and proceeded to slice my thumb real good. Trust me, you will only do that once.
Well happy "bringing an axe back to life" project to you. It is a rewarding feeling when done. Next step is then to get outside and go use that baby for 12 hours a day, non stop, except for lunch.
Good luck!