Many methods of making things used by alchemists actually came from industry. This is one of them. Or maybe it was invented by alchemists first and taken over by industry, I don’t know just now and haven’t come across much research on the topic. (that’s what I like about this, there’s wide open spaces to explores)
Either way, it is very simple. The method I am following is from the Libellus de Alchimia by Psuedo-Albertus Magnus.
The text is probably later 16th century, but originated in the 14th, with some changes and accretions over time.

So, first take your ashes, either putrid oak ashes or burnt lees of wine, (which were apparently use for washing garments), grind them finely and add a 6th part of quicklime, that is CaO. Mix together and place in a woven cloth over a jar or beaker.

Like this:
potassium hydroxide making washing the ashes

Pour boiling water into and through it, until it has finished reacting (The CaO reacts with water to form CaOH and such) and you think enough salts have been washed through to the liquid beneath.
Then put the filtrate into a vessel and distill by filter, as per the photo at the top of my page, although this is the setup I used this time:

KOH making distillation by filter

The instructions say to heat in a fire until the solution evaporates and does not fume. I haven’t done that, just heated it until it is dry, and here it is:
KoH moisture nearly gone

If you heat it until it stops fuming it goes like this:
KOH dried

I don’t know why it is brown, maybe there are some water soluble impurities. It probably isn’t 99% KOH anyway.

What you should have made, with the addition of the lime, is KOH, from the KCO3 present in the ashes. That is, potassium hydroxide from potassium carbonate. You’ll be familiar with sodium hydroxide as drain cleaner or such. When you dip your fingers in a solution of KOH or NaOH and rub them together it feels slippery like soap. That’s because the potassium hydroxide is reacting with the fats in your skin to make soap. In fact that’s how you make soap, mix the carbonate or hydroxide with a fat, or an oil. Apparently it is quite smelly, I aught to try it some time.
Of course the difficulty is that the proportion of KOH you end up with depends in part upon how much of the carbonate is in the original ashes.
The final step in the Libellus de Alchimia is to melt the hydroxide in a furnace, which should ensure there is no water left within it. Left by itself it will absorb moisture from the atmosphere. It should melt at 406 degrees C, but really there’s not much need to measure that.

Then you just need to use it in your alchemy. I’m still trying to work out exactly how, the texts aren’t that much help, yet the Libellus is insistent that it is of some importance in the work.

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