Buy inexpensive vodka or grain alcohol that has a high number of distillations. The type of "neutral spirit" you'll start off with depends on what kind of liquor you want to make; in other words, what kind of essence did you buy? I'll be using and continuously drinking vodka, so I'll just refer to all liquor as vodka from now on.
Fill each cask fully with hot water in order to let the wood expand, creating a better seal. This is especially important when using new casks or casks that haven't been used in a long time. Let the casks sit for 20-30 minutes and then dump the water out.
Fill each cask more than halfway with vodka. Let sit for 30 minutes; then, rotate 180 degrees and let sit for another 30 minutes. This will sterilize the inside of your cask! When finished, dump the vodka into your mouth, sink, or friendly hobo.
In a large seal able container (such as a clean soda or milk bottle), mix one bottle of liquor essence, 750mL of vodka, and one ounce of glycerin. Finally, add the amount of sugar recommended on the essence bottle. Close the container and shake until all sugar is dissolved.
I don't have a photo for this step since I looked terrifying and decided not to add it. However, a vigorous 90 second shake will do the trick.
Pour the combined ingredients into your cask. I recommend using a clean funnel (not one covered in motor oil).
Like a fine wine, liquors from essence improve with age! However, you do lose a small amount from using an actual cask (the angels share in whiskey production) so if you age it for more than two or three months more will be lost.
Also taste it regularly as the longer the beverage site the more flavor it will gather from the cask.
Your product will be drinkable in a day or two but will only improve with age. I left mine in for three weeks and it turned out amazingly.
Drink and repeat. Looking for the next great home brewing project? Try DIY mead making -- it's equally as lazy a process and is easier than brewing beer!