How do I know if the food I’m giving my cat is making him sick?

I would definitely monitor things like your cats stool. Sometimes if it is not the right nutrients for him, sometimes the stool can just be a little, not as solid. Also, it can be really foul smelling. Also if your cat's energy level decreases at all, or if the coat does not look as beautiful as before you started that diet. I think at that point, it's a good idea to chat with your veterinarian about switching diets.

The most common thing we see is food allergies. If the allergy is consistent 365 days a year, then it's probably not fleas and it's probably not seasonal. I would approach illnesses that are related to food the same way.
If it's something that you're feeding—maybe it's a new additive that you added to the cat's diet or a new ingredient or perhaps even a new bag of food—and right at that point you see that your cat has been constantly ill with a variety of signs, it's a strong possibility that it's the food. If it began with the initiation of a new ingredient and it's consistent after the cat has been eating the new food, that is kind of a big red flag for me.