You may begin curing the citizens of their slight ailments when you take up residency at Pembroke, though some might have a heavier sickness on them than others. To start, you will only be able to cure level one sicknesses (fatigue, anemia, and sepsis), though you might have a harder time collecting the enzymes to create the later versions compared to the very first one. To expand your knowledge of cures, there are two specific locations to look out for.
For Level 2 cures, when you are sent to the morgue to locate supplies for a medicine. This will eventually lead you to fighting a Blinker Skal (a Skal that can shift away/around). After the fight, you can collect an enigmatic formula that, when analyzed at a workbench (and there is one nearby to use), you'll unlock the Tier 2 cures (cold, bronchitis, pneumonia).
For Level 3 cures, when you proceed to Whitechapel and eventually have to go to the cemetery, proceed to the latter location from the former (from the church in Whitechapel towards the cemetery). Eventually, you'll come across an open gate leading to a closed gate right into the cemetery. Before you enter the cemetery gates, turn right from here and you should see a corpse in a depression in the ground. Looting this corpse will give you the formula for the Tier 3 cures (headache, migraine, neuralgia).
You can tell what someone has by using Sense and reading their Blood Quality, or by holding R2 to bring up their quality in dialogue. If you're not near them, you can bring it up in the Citizen List in the menu, which will only update if you've rested. To get this trophy, you simply have to cure ten ailments off of citizens. Even if it's the same citizen being infected repeatedly, as long as you have administered the cure and they go from being sick to Recovering, it will count. In the long run, you might be better off curing several before sleeping. If you wait too long, it will graduate from a Tier 1 to 2 and 3, bringing the status of the district down and making it harder to cure.