I am familiar with the axioms of ZF set theory and some basic uses of them to completely formally construct more complex objects such as natural numbers etc. However I have pretty much no background in formal logic.
What I'm looking for is a book, or a collection of books, that would lead me through all the necessary material up to and including the proof of independence of axiom of choice from the other axioms, preferably also with some further insight into its weaker forms such as the axioms of countable/dependent choice.
Could you recommend something?
Definitely the book for this is Jech The Axiom of Choice. The book includes all the exposition necessary, from the axioms of set theory, permutation models and forcing, up to weaker choice principles such as $\sf DC$.
(I will say that I am not a huge fan of the approach using atoms, but it is a good way to get the basic idea if you're not very familiar with forcing.)
You can find basic information in Jech's Set Theory book, in particular Chapter 15.
Other books which include information relevant for your question are:
Not quite to your request, but useful nonetheless are:
The former deals with implications of the failure of choice in mathematics, the latter deals with historical review of choice principles and include lists of choice principles.