Short answer, a Medieval doctor of our world be unlikely to be able to do what you're looking asking.
Medical experimentation and autopsies were not an accepted part of culture until much later than this historically in our world. However, that doesn't mean that you can't build a world where this is possible, there will just have to be specific changes in culture and science made.
In the late 1400s in Padua and Bologna, Italy, the sites of the world's first medical schools, Pope Sixtus the IV issued an edict permitting dissection of the human body by medical students. Before such edicts from religious leaders, it was considered a crime to dissect the human body and criminal prosecutions for "body snatching" by students of anatomy date back to the early 1300s.
By the 1500s, the autopsy was generally accepted by the Catholic Church, marking the way for an accepted systematic approach for the study of human pathology. While a number of "giants" around this time, such as Vesalius (1514-1564), Pare (1510-1590), Lancisi (1654- 1720), and Boerhaave (1668-1738) advanced the autopsy, it is Giovanni Bathista Morgagni (1682-1771) who has been considered the first great autopsist. SOURCE
If you're talking the 1400s (which is the 15th century) there just hadn't been enough experimentation in order to really do what you're talking about. You'd need hundreds of years of acceptance and schools to get to a place where someone might be able to accurately declare the cause of death.
And what you're talking about is actually forensics. And chemistry. By the 1600-1700s, we'd started to get a grasp on chemistry, a bit, but we were not at the level of forensics.
This practice was actually born later, in the Victorian Age, when the collective observations of hundreds of doctors and students through the ages, plus a start to understanding chemistry gave it a boost. It was nowhere near as good as it is today, but this was the start. See this article on how they first figured out arsenic poisoning in the 1800s. You'll notice that observational experimentation and the recording of those results, along with knowledge of chemistry was used.
So your proto-pathologist would have to be, gosh, 300-400 years more advanced than everyone else, if you are using the real world as a basis. Even if he's a genius, keep in mind that he cannot have seen everything, and pathologists, of this time and that relied on books published on the subject, the experimentation and observation of others.
The answer here is that you are going to need to change your world a bit. It has to have been a) acceptable for many years (perhaps hundreds) to cut into dead people as a method of learning b) there have to be books or sources of information from which to learn c) there has to be a base scientific knowledge of chemistry, even if it isn't fully understood.
In the Victorian era even, sharing information of this sort did not happen quickly, but you will notice a spate of landmark cases from about 1800-1900 as the science was born.
A person you might want to look at is: Alexandre Lacassagne from France. Don't look so much at the psyche work he did. There's a lot of bunk there, although he is better than his Italian counterpart, Cesare Lombroso, who was big into equating physical body types with a predisposition for criminality. Still, it would be interesting to have a protagonist who is right about forensics but attributes criminality to something crazy like phrenology or an imbalance of the humors.
You say that:
Poison is a combination of plant, venom, and minerals, so symptoms would be difficult to diagnose.
One of these can easily cloud the symptoms of the other, but if the combo is known--that can help. Some poisons have physical symptoms attached to them that could be easy to see--it's up to you as the author to attribute them--so the pupils normally dilate pretty quickly after death--but one of your poisons might prevent that for a few hours or it could be something like the rosy glow present in bodies that have died of carbon monoxide poisoning, or any number of purely physical tells, like body positioning as a result of muscle convulsions, that sort of thing. The quicker they get hold of the body, the more likely they can start to come to conclusions (except in the case of some poisonings, a symptom of which can be body preservation).
Your proto-forensics guy might understand that when a certain compound is present, it turns blue when mixed with urine/ammonia. He might not know why, but he could know that it's true.
The wound might be obvious or small, but an inspection of the body would reveal it. Then, he would perhaps dissect the area and test the tissue, looking for anything out of place. A great help would be the invention of the compound microscope or something very like it, although you might be able to get away with an efficient magnifier.
The Maesters of Game of Thrones are a pretty good guide to upping science patchily. They have a repository of knowledge, and "earn their chains" by studying certain sciences. In this way, RR Martian has been able to slip in things a little beyond the day as part of the world (which is different from ours) and you can do the same.