The Healer trophy/achievement is awarded if you use Medicine (bought from the doctor in town) on a sick character.
In order for a character to become sick, your relationship level with him/her must be at Lovers or higher. After that, whether the character becomes sick or not is random. It might take several years for this to occur, so the earlier you can get up to a high relationship level, the better. The character may also never get sick--such is the case with randomness.
Once the character is sick, talk to him/her and give him/her the Medicine, just as you would Berries/Coppers/Rabbits/Rice. You should immediately receive the trophy/achievement.