Most homesteaders have somewhere an axe with a broken handle. It's a fun project to make your own axe handle that is often better than the ones out of the shop. In the shop the grain of the wooden handle is often not straight. This is because it's made by a machine. The grain should run to the point of the handle. If you make it yourself than it's the red oval part that I marked.
What type of axe head do you have?
There are many different types of axes from survival to splitting axes. All need different handles. So research which length should be correct for your axe head, use and your body size. Don't sharpen the axe head keep it dull till you've fitted the handle. Otherwise you cut yourself later in the process.
Which wood?
Most dried tough wood is fine for short handles for longer handles you need to have hickory, ash or oak. The best wood is dried inside where the humidity is lower than outside. I've chosen Robinia Pseudoacacia for my short axe handle. It's the only wood that I have available. It's very hard but doesn't have much flex. Split the log with a axe and some wedges. This is better than sawing. With splitting you split across the grain what result in less fibers that are broken.
Now you take the best half and draw roughly the shape of a handle on it. The orange drawing follows the grain but I made it myself easier. It's not a problem for a short handle to not have perfect straight grain.
Tools needed
If you got an axe and a saw than that's enough to make an axe handle. A knive, draw knive, bandsaw, spokeshave, angle grinder and sandpaper can make the job much easier and faster.
Fitting the head
If you look at a axe handle you notice that there are 2 different openings. The smallest opening should be the bottom and the biggest opening should go to the top of the handle. Don't mix them up otherwise your handle will never fit.
When the handle looks like it could fit you take the axe head and you check where the handle is to big. With a knife or a rasp you take that part off till the axe head perfectly fits the handle.
Now you take a saw and you saw on the green line. If you have a wooden wedge than saw with a normal saw if your wedge is from iron than use a thin metal saw.
Now fit the head for the last time check you have the correct side of the axe head up and insert the wedge. A wooden wedge can use some wood glue before you insert it. Now hit it with a hammer till it stops going deeper. Don't hit to hard you will break the wedge.
Finishing
Finish the handle with some sanding till it smooth. Not to smooth you need the grip. This is also why you never paint or lacquer a handle. I give the handle a coat of wood oil this will tighten the handle a bit more. Keep the axe dry but outside were the humidity is higher. You don't want the wood to shrink.
Maintenance
An axe doesn't need maintenance, but it requires proper handling. Don't hit the axe head with a hammer, don't use it as a hammer, don't use it as a lever and don't let it sit in the rain or sun all day long. The only exception is a splitting maul that type of axe can often be used as a sledge hammer.
Last but not least. Check the axe head before you swing if it's loose it's dangerous to use.