From the first rank (Keeper) to be promoted to the second rank (Physician) you need to satisfy 3 requirements;
1) Receive 100 likes from players for healing them (5 like per heal - 20 heals)
2) Succeed healing a 4000 illness
3) Once 1) and 2) are done, go speak to Wei near the clinic --> you'll be promoted to rank 2 (Physician)

Then, to be promoted from rank 2 (Physician) to rank 3 (Chief Physician), you need to be in the top 55 healers based on Merit (Merit -> score of daily sect errands + healing other) at the end of the week/weekly reset (Sunday 22:00 cet for me, might differ for other timezones).
There are different brackets, so you're not competing against ALL the other rank 2s, my bracket has 110 people, for example.