How do I leverage this to make biochar, and then how do I use the biochar to improve the vegetable patch ?
Pack the hole with split logs standing on end.
Build a rip roaring fire on top. Let burn for about an hour.
Put fire out, you should have full sized logs that are very black.
Pile compost on top and soak with water. The fire will be trying to restart in the logs for some time, prevent this from happening.
Let pile sit for a year and you can keep adding on top of it.
Dig compost off top and set aside.
Mash charred logs with sledge hammer and then mix in your compost. Add to garden. Repeat.
The charcoal acts like a sponge for water soluble nutrients in the soil; if you add it to the soil directly, it will suck them out of the soil first, temporarily depleting it. Long term it acts as a reservoir, but you want it to go into your garden full.
As for the garden organic waste one can tie itself in knots trying to balance this or that. In my experience shallow trench composting works just fine. You do not get a smelly pile leaking fluids, packs of slugs / swarms of flies. No extra work (mixing, distribution). Earthworms will also thank you. You may have issue with hardy weeds and rats, so you milage may vary. One can ferment the weeds in a plastic bag before buring them lowering the chance of spouting, but this is extra and quite smelly work. One can also dig deeper(+30cm of earth cover). Less chance of weeds and animals digging to snack on a rotten carrot.