There is no specific book that will teach you how to diagnose illnesses. First of all, medicine is an extremely large and complex science, consisting of different specialties, and every one of them has its own large spectrum of knowledge to study (eg. your question - skin conditions are subject of dermatology, flu - infectious diseases etc.). Doctors go through the basics in all fields during their ~6 years studies, and later become specialists only in a particular branch.
Secondly, just knowing the symptoms of a disease won't qualify you for diagnosing. It's a complex process, that involves a set of special skills, such as knowing how to examine a patient, understanding imaging/lab, and so on.
And here comes an important note: most of what I described is not only taught on a basis of medical literature, but also by a mentor who not only is a good specialist themselves, but also knows how to make other people capable of using all this knowledge.
Last but not least, the comment regarding background knowledge is also completely true and is to be taken into account.
Having said all that, I assume that your goal is to get a peek inside how diagnosis works, in this case i'd recommend some "Introduction to..." textbooks for different medical specialties, they might be a bit challenging to understand without required theoretical background (eg. histology, physiology etc.) but will give a nice overview of what do those specialties work with.