The bad news is, there doesn't seem to be an easy trick to it.
The most important thing I've found is that you basically only get one chance to get the cord through the holes while it has a decent tip on it.
Most of the replacement cords come out of the package with at least one end having a decent tip where most of the fibers are stuck together. If only one end has a good tip, plan your install to stick that end through both the cocking block and the spool.
If neither end is very good, then I've had good luck using a little super glue or fletching glue to bond the fibers and make it a little hard. Don't use so much glue that you increase the diameter of the end as it won't fit through the holes.
I have found that using a lighter to try to melt the fibers together does not work.
I carefully start the end of the cord into the hole and twist in one direction while I push the cord through. I usually can't get it far enough by hand, so I then use a needle-nose pliers to twist and push some more. I get enough of the end through to grab the end with the pliers and pull through the rest of the way.
Don't pull the end unless you can grab the entire end with the pliers. If you only get part of it, only that part will come and the rest will be stuck and you'll have to back it out and start over. If the entire end doesn't come through on your first try, back it out and glue the end and start over.
If the end is in rough shape after your first try, you can cut a little off and glue a new end and try again.