that's a really fun job and a pleasant way to spend an afternoon. it would really help to have a model to go by. usually this is the old axe handle from the same head.
here's my process:
working from a clear log (i use black ash because that's what we have) a bit longer than your desired handle
split it in half, starting with an axe driven by a sledge, then a maul in the crack opened up
then you get to see the grain
quarter the halves
maybe one of the quarters has a nice curve to the grain that can be incorporated into the handle- pick that one
split the quarter to a 2" board
even it out with a sharp axe
lay out the axe handle shape on the board
rough it out with the axe
smooth it with a drawknife
take it from square to octagonal with the drawknife
then round it out nice with a spokeshave
maybe a rasp for the knob if you like it round, or just saw across the swell at the end of the handle. both look nice imho
spokeshave the other end of the handle to fit in the eye of the axe
saw a cut in the end of the handle for a wedge
bang the head onto the handle using the weight of the head
put a hardwood wedge in as far as it will go
put a metal wedge in diagonalwise
oil the handle
start a fire with the trimmings
i think a split handle that follows the grain is far preferable to a cut one- stronger. the handles on most store-bought axes, even decent brands like an oxehead, are pretty poor. this sounds like lamenting the good old days- and i wasn't even there!