Simmer the water, brown sugar, and white sugar in a saucepan over medium heat for 5 minutes.
Remove from the heat and add instant coffee. Stir thoroughly and let the mixture cool.
Pour the mixture into your jar and add the vodka or rum and vanilla extract. Stir thoroughly or give the jar a good shake.
You can drink your coffee liqueur right away, but the flavors will improve if you let it sit. Leave your coffee liqueur in a dark, room temperature space for 10 days to two weeks.