This is the same goal as the Eradication! achievement, only to eradicate 2 diseases. Similarly, the Medic is the best way to approach clearing cubes once a disease is cured since they automatically treat diseases in cities without expending an action.
If you are struggling with this achievement, I recommend trying to play with fewer characters. Fewer characters lets each character have more turns, i.e. the Medic has more turns to clear cubes.