The speculation about Mongols and the like is all very well, but the problem is with the documentation: it simply does not exist, as far as I know, for individual Mongol warriors.
There is a very limited set of people who will meet the OP’s requirements: (1) documented; (2) did not kill with firearms; and (3) not an executioner.
If you stab a guy, his wound gets infected, then the infection kills him… Would that count?
For most bows & arrows, in most armies (the kinds that are mostly infantry), true. For steppe horse archer type armies, like the Mongols (or, for that matter, the Parthians that made Crassus look like such a doofus at Carrhae, or the Huns rampaging through Late Antiquity): Those guys are indeed turning you and your buddies into pincushions. And then, when you try to complain, they ride away before you can catch them, all the while gleefully farting in your general direction.
OK, the farting part isn’t historically attested. But that’s how I picture it.
Although, obviously, much of the destruction wrought by the Mongols involved taking of cities and massacring civilians. Presumably, they got off their horses, and picked up siege engines and stabbing/beheading type tools, for those parts.
Everybody’s discussing ancient combat, assuming that the killer mentioned in the OP was some professional soldier.
But I’d like to ask about something that happened in our own times:–the massacre in Rwanda in 1994.
A million dead, mostly hacked to death by “average”, anonymous people.
It’s even more horrific when you think that it happened in modern times and was documented in real time, with the full knowlege of every political leader and potentially every news camera in the world.
It may be possible to actually find a specific person who would “win” the title that the thread title is looking for: the human who has killed the most people by hand.
He might even be sitting right now with some buddies over a beer --and bragging about his feat of killing.
It was slightly more complex than that.
It is true that the Mongols made extensive use of light cavalry of the “shoot at you and fart in your general direction while trotting away” variety.
They also, though, had heavy cavalry, who used hand weapons to kill the shit out of people. The ratio was supposedly around 60% light, 40% heavy.
In addition, by invading urbanized bits of what is now North China, and recruiting all sorts of Chinese specialists, and by invading cities along the Silk Road, they also had Chinese and Islamic siege engineers, with all the latest in siege goodies.
This was what made them so terrifying - they were a true “combined arms” army, who did not suffer from the weaknesses of a force made up of only one type of troop.
They would typically use well-directed tactics to destroy enemy armies: send in the light cavalry to shoot 'em up; stage one or more ‘false retreats’ to get 'em to chase; then, with their numbers whittled down and their formations disrupted - send in the heavies to chop 'em up and to cause panic; then the lights again, for the pursuit until they were utterly destroyed.
Probably - Mongol troopers closed when they had to, with lancers used particularly to break disrupted formations after the arrow hails had done their work and if they hadn’t been sufficient in of itself. But the bow was always the primary weapon and probably did the most damage most of the time. Even if we allow for the heaviest casualties during a rout where lancers could ride down and skewer folks, only a smaller portion of the army were lancers. A majority would probably have pursued and shot you down. For example Mohi/Sajo River the retreating/routing Hungarians were funneled through a gauntlet when a gap was deliberately left open as a trap - with mounted archers loping along either side of them, shooting them down.
Conversely the lightly-armored troopers on their small, passive ponies ( mostly mares and trained to be responsive, rather than to fight ) were vulnerable to more heavily armored European knights on their larger, battle-trained mounts ( almost always stallions, trained to bite and kick ). They could took heavy casualties when forced into hand-to-hand combat, as in an earlier stage of the fight at Mohi. So they quite naturally preferred to engage in close combat only if the situation demanded it. Much of the damage they inflicted came at a distance with arrows and artillery.
ETA: Since Malthus beat me in to mention ‘heavies’, it is worth noting that a Mongol ‘heavy’ was still lighter than any European knight ;). Heavy cavalry is probably in this case a misnomer. The were relatively more heavily armored, but only relatively.
Plenty of examples of that in ancient times too unfortunately:( The OP was about combat, not massacres of civilians or executions.
So assuming this era is the correct answer. Which actual person do we know from this period who spent their lives fighting on the frontline for Caesar and Augustus? Lucius Vorenus is only name I could think of (and he is only mention once as fighting in the Gallic Wars), but I am sure there must be more. The settlement of veterans was a massive political issue in that era, and we have grave markers etc, which generally mention military exploits above all else.
Sure; but then, the heavies were only used in a small part of the standard Mongol battle plan - to break the enemy’s resistance (after they were already worn down by the lights). The lights would them be used, again, to harry and destroy the panicked troops on their retreat/flight, by pacing alongside and shooting 'em down.
In short, the heavies had a small, but very important, role to play in the Mongol battle plan; probably the actual number of casualties caused by them with ‘hand weapons’ was small … far more significant was their moral effect, in causing an already weakened and/or strung-out enemy to panic (though this is often the case with heavy cavalry).
Obviously, the Mongols resisted ever having a direct head to head confrontation between their heavies and (say) unwinded, unbroken European Knights - a “fair fight” is the last thing a Mongol went for. 
Rather, they would use their light cavalry to torment the knights and, if possible, draw them away from their infantry; if the knights sat still - good, the Mongols would use 'em as target practice; if, as was usually the case, the Knights charged - why, then the Mongols would retreat - all the while shooting at 'em - until the knights were fatally weakened and strung out: then crush 'em with the heavies (well rested and waiting some distance away). Usually, the sight of their heavy cavalry being destroyed was enough to cause the infantry to run, whereupon they would be chased and destroyed in turn.
If we are going by that sort of metric, the Aztec priests conducted hundreds of human sacrifices every week.
This guy probably holds the record in that respect.
Ugh, apparently he put a bullet into the skull of three every minute, and only did his job at night so the bodies could be disposed of more easily.