First, you will need a barrel and a melee weapon (optional).
The easiest way to obtain both of these is by going to the Training Grounds (host a private lobby and look in the top left).
There's about 11 barrels in the spawn room you can abuse to your hearts content, and theres a flamberge right outside the door leading to the open environment.
To start, pick up a barrel and your flamberge, and position them as pictured.
Now sprint jump and attack shortly after, and you should be sent flying!
Congratulations, you've now learned to Barrel Punch™!
Position yourself like pictured, then throw the barrel, jump, sprint, and slide, and try to land UNDER the barrel as its going forward to Barrel Boost™.
This one takes a lot more effort to pull off and is very inconsistent, but it lets you get high speed with just a barrel.
In the vast majority of maps, I'm pretty sure Barrel Punching™ is literally useless, but you might find some utility in the Barrel Boost™ instead, but its way harder to pull off and is probably not worth it 99% of the time. I'm hoping for workshop maps to be a thing so random niche speed tech like this can actually be viable in some kind of environment, but I'm not holding my breath.
Enjoy your niche movement tech!