This is the recipe I use to make Kahlua.
1 qt water
2 1/2 cups Sugar
3 tablespoons of instant coffee
1 tablespoon of Vanilla
2 1/2 cups Vodka
Bring water, sugar, and coffee to a boil in a saucepan. Simmer VERY slowly for 3 hours. Mixture will be very dark and syrupy. Cool. Add vanilla and vodka. Makes 7 cups.
Options:
Instead of using instant coffee, make a 1 quart batch of coffee to whatever strength you prefer, then use that as a replacement for the 1 qt of water and 3 tablespoons of instant coffee.
As you probably noticed there is no brewing involved, except for making some coffee. The alcohol comes from the vodka and sweet taste comes from the 2 and half cups of sugar. Effectively, Kahlua is just coffee flavored syrup with some vodka (or other source of alcohol).
It would be possible to follow a process similar to making Port, where you ferment a must (brewed coffee and some source of sugar), allow the fermentation to proceed, stop the fermentation process at some point, then add some source of sugar back to provide some additional sweetness. However, the end result would probably be nothing like Kahlua, though it might make an interesting experiment.