TL;DR: 6 Levels are enough
If you dislike math and only want to know how it works at level 6, you can skip to part 3.) right away.
1.) A Fighter 2 / Monk X strategy requires a Character Level of 12 to achieve superhuman speed.
Take Mobile as a Human Variant Starting Feat, take two Fighter Levels to get access to Action Surge, then take Monk Levels for Unarmored Movement. You can then use Action Surge and Step of the Wind to Dash three times in one round, which gives a total movement of
$$ (6+2+k)\cdot4 $$
squares where \$6\$ is the Human base speed, \$2\$ comes from the Mobile feat and \$k\$ from Unarmored Movement (the amount of which we leave variable in order to determine the least amount needed to break the world record). A round is \$6\$ seconds long and a square corresponds to \$5\$ feet which is \$1.524\$ metres. Hence, even if you have the maximum of \$k=6\$, this will only take you
$$ (6+2+6)\cdot4\cdot1.524\approx85.344 $$
metres in one round. Thus, we need another round to cover the remaining
$$ 100-((6+2+k)\cdot4)\cdot1.524 $$
metres. In this round however, Action Surge will no longer be available, leaving us with
$$ (6+2+k)\cdot3 $$
squares of movement, which is a speed of
$$\frac{((6+2+k)\cdot3)\cdot1.524}{6}$$
metres per second. Putting things together, this strategy lets you run \$100\$ metres in
$$t(k)=6+\frac{100-((6+2+k)\cdot4)\cdot1.524}{\frac{((6+2+k)\cdot3)\cdot1.524}{6}}$$
seconds. Plugging in the possible values of \$k\$ yields
$$ t(2)\approx 11.123, \quad t(3)\approx 9.93, \quad t(4)\approx 8.936, \quad t(5)\approx 8.095, \quad t(6)\approx 7.374.$$
Hence, a Monk 9 / Fighter 2 is still quite a bit away from beating the world record, while gaining another Monk Level already makes them superhuman at a total Character Level of 12.
2.) A Fighter 2 / Barbarian 3 / Monk 2 does the same at Level 7, provided that they can find a way to sustain their rage.
What follows is a more detailed and precise elaboration of the idea suggested by user48255. Again, take Mobile as a Human Variant Starting Feat, take two Fighter Levels to get access to Action Surge, take three Barbarian Levels to get access to the Elk Totem Spirit, then take two Monk Levels for Unarmored Movement. While raging, you can then use Action Surge and Step of the Wind to Dash three times in one round, which gives a total movement of
$$ (6+2+3+2)\cdot4 = 52$$
where \$6\$ is the Human base speed, the first \$2\$ comes from the Mobile feat, \$3\$ is the Elk bonus and the second \$2\$ is the Unarmored Movement. Again, the second round without Action Surge allows only
$$ (6+2+3+2)\cdot3 = 39.$$
Thus, this strategy lets you run \$100m\$ in
$$6+\frac{100-52\cdot1.524}{\frac{39\cdot1.524}{6}}\approx 8.095$$
seconds. You do have to take care of sustaining your rage, though. Maybe put a d4 on the floor and step on it. ;-)
3.) But Monk 2 / Fighter 2 / Wizard 2 does the job already at Level 6, provided that a human is allowed to become a Bladesinger.
The next build follows an idea that was expressed by the user Benjamin Olson in the comments. Again, take Mobile as a Human Variant Starting Feat and take two Fighter Levels to get access to Action Surge, but then add two Wizard Levels for Bladesong (assuming the campaign relaxes the racial requirement) before taking two Monk Levels for Unarmored Movement. Note that just like Ki Features or the Totem Spirit, Bladesong itself does not count as magic for gameplay mechanics. Using Action Surge and Step of the Wind to Dash three times in one round while Bladesong is active gives a total movement of
$$ (6+2+2+2)\cdot4 = 48 $$
squares, where \$6\$ is the Human base speed, and the \$2\$'s come from the Mobile feat, the Bladesong and the Unarmored Movement respectively. In the second round, without Action Surge, this is reduced to
$$ (6+2+2+2)\cdot3 = 36$$
squares of movement. Hence, this strategy lets you run \$100m\$ in
$$6+\frac{100-48\cdot1.524}{6\cdot1.524}\approx8.936$$
seconds and thus achieve superhuman speed at a Character Level of 6.