What is the quickest way to kill someone?

depending on how limited your resources are, or how much time you have, or how big of a psycological advantage you have over the other person; the answer to that question can change.

But, under any circumstance i think its best to aim for the nose with either the palm of your hand until they began to look away or flinch, then get them on the ground with you above them, then alternate between the nose and throat.Once they've suffered a severe beating and you know you have enough time to turn your back, grab a big rock... or choke them out with your belt or something.

Hand to hand: put your thumb in their eye, their brain hemorrhages and they die.

With a weapon:

Blade: Stab at the chest twist pull out. Or slash at the throat, but that takes skill.

Blunt hit to the head, crush their head when their down.

Gun: If I have to explain this is a head shot weapon.