This page contains a detailed description of Hippocratic Oath available in Vampyr. You get it by healing 10 citizens with medicaments. Below you can find the most important information on how to get this trophy.
Use sense mode (detective mode) to check a character's health status or examine an NPC during a conversation. At the beginning of the game most of the characters are fully healed but night after night, the citizens, even the healthiest ones, can start to get sick.
There are 9 diseases in the game - a character can suffer from pneumonia, have a headache or simply be tired. Each of the diseases have a specific medicine that can cure them (Jonathan can prepare them in a safehouse). Simply approach a desk and create a medicine (the picture above). Each medicine requires certain crafting materials which can be found in the world. You can buy them, find or dismantle useless items.
Take a medicine to a patient, ask about their health status and give them a medicine (the picture above shows an example of that). This gives you a little bit of XP and improves blood quality. That patient will get better the next night.
To unlock this trophy you have to cure 10 citizens. You don't have to try to get it right at the beginning of the game and wait until someone gets sick. However, you have to remember that free exploration of the world is available until you initiate the fight with The Disaster, the game's final boss, during the sixth chapter.