Humane treatment of animals is a rather modern concept (which doesn't mean deliberate cruelty is desirable).
There are several more or less conflicting goals here:
- Kill it with as little suffering as possible: Should probably be done by getting it unconscious as quickly as possible, either through a hit to the head (that actually hits the head), or massive bleeding. Once unconscious it probably doesn't feel anything.
- Get as good quality fur as possible: Blunt attacks to the head. Misses will cause less damage to the fur than stabbing/slashing does. It probably takes longer than using something that also causes bleeding, though.
- Get as much weapon skill training as possible: blunt attacks from the weapon you want to train, aiming for the head. You should probably use as light a weapon in that category to get as many attacks as possible before the animal dies.
- Also note that aimed melee attacks automatically hit the targeted body part when the target is unconscious. This is not the case with missiles.
The quality of a fur depends on:
- The quality of the carcass. If damaged too much one or several skin qualities cannot be achieved, and a low quality carcass probably means a lower chance to get good results within the possible range.
- Your character's effective skill, i.e. Hideworking with subtractions from fatigue and encumbrance. Drop everything carried and wait until the fatigue is zero if trying to get as good a result as possible.
- Tool type and quality. A stone knife limits the maximum possible skin quality, while a broad knife adds a bonus. Higher quality tools provides a higher bonus. I'm not sure I remember correctly, but I think a normal quality broad knife allows for a masterworks result, while any other kind of normal knife (i.e. not a stone knife) has to be at least fine.
Thus, your skin can be of a poor quality because of a carcass in poor condition, low effective skill, or poor/unsuitable tools. Also note that each step of the process runs the risk of getting a poorer outcome that you could have gotten, degrading the quality of the result. However, I think a sufficient skill will prevent the worst outcomes unless the tools/input items drag it down.