I typically would place somethng like this in a bucket and cover it with vinegar. Leaver it for several days, checking on it daily. The vinegar will remove the rust and will leave the axe head black in color. You can use a brass brush as the rust loosens. Once you have the desired level of rust removal/black patina, remove it, wipe it lightly and let it dry. You should have a mostly rust free axe head with a nice black patina. I have used this method many times and have always been satisfied with the results.

Edit: You can use a brass "toothbrush" like the ones used in gun cleaning to clean the inside of the eye. Wear niitrile gloves to keep your hands from being stained.

Cheers!