Yes, yes there are easier ways to split wood, but the OP wasn't asking about what to use to split wood, they asked how to sharpen what they already have. I have a fireplace inside and a pit outside. I don't split more than a cord of wood a year and I can't justify spending money on something dedicated to splitting wood. Plus, I really like using an axe a few days a year.
I use a Gerber (Fiskars) splitting axe. It's like half axe, half maul. This is the new version.
http://www.gerbergear.com/Cutting-tools/Axes/36-Power-Splitting-Axe_31-003636
I also have an Estwing Camp Axe.
http://www.estwing.com/ao_campers_axe_long_handle.php
To keep them sharp I use a file to take any big nicks out of the blade. After that I use a rough sharpening stone and just follow the profile already ground into the blade by the factory. It's an axe, so I'm not worried about it being perfect and I don't use my fine grit stones on it. Something that you can hold in your hand works to get the angle correct. It's a lot easier to clamp the axe in a vise or set it on your lap and take the stone to it, than the other way around. Lots of people rave about the sharpening pucks available for this purpose (
https://www.amazon.com/Lansky-3289-Dual-Grit-Sharpener/dp/B017T2GG5E), but I didn't see a reason to buy something new when my existing stones work fine for me. YMMV and I'm sure there's a proper way to do it, but this is how I do it.