Typically the grain should be running vertically in an Axe or hammer handle. The shape looks well done. Good fit in the eye?
"the grain should be running vertically " lost me on that one... 
Vertical in line with the orientation of the blade. The handle shown appears to have the grain running across. The sides of the handle should look like a bow belly from a conventional stave.
Most commercial handles only loosely adhere to this feature.
By "grain" I am just talking about the orientation of the growth rings.