I am installing an under the sink water filter that needs a constant power source. The only outlet available under the sink is a single 2 outlet box wired to a switch above the counter for the garbage disposal. My question is how can I make one of the outlets hot for the filter all the time and the other outlet still operated by the switch for the disposal?

Since the outlet under the sink is completely controlled by the switch above, there are two methods of getting constant power to a receptacle for the filter.

One is to install a new 3 wire cable from the counter switch to the duplex receptacle below, switching one half with the red, and having the other half constant power with the black.

This may not be so easy to do, so, assuming you have a dishwasher next to the sink, you could install a new duplex under the sink and run a short piece of cable and connect into the DW junction box.