Find a glass bottle or jar big enough to hold the liquor (at least 3 cups). Pour in 2 cups of rum (or vodka), the vanilla, and ground coffee. Shake to combine and keep on the kitchen counter for 2 to 3 days to infuse the liquor with coffee, and shake every so often to redistribute the ingrediants. Note: your results will be better if you use freshly ground coffee beans.
Strain the liquor through a fine mesh strainer, then through coffee filters until all the grounds are removed. Throw away the used grounds.
Make a simple syrup by placing the water, brown sugar, and white sugar into a large sauce pan. Simmer over medium heat, stirring frequently for 5 minutes or until all the sugar is dissolved. Remove from heat and let syrup cool in the pan to room temperature.
Combine syrup with infused, strained, liquor and pour into individual glass jars or bottles, and keep in the refrigerator or dark cupboard for a couple weeks. At first there will be a strong taste of alcohol which will mellow in 2-3 weeks, resulting in a smooth, caramel, coffee liqueur.