The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference to the topic, using a more modern language (schemes etc.)?
Have you looked at
Thesis of Peter Toth (follows Deligne's approach)
Kerz's articles (amazing innovations in class field theory by Kerz and Wiesend); there is also an excellent Seminaire Bourbaki expose on this by Szamuely (in French).
There are many other good references, but hope this can help.
1) Our (slightly pseudonymous!) friend, Brian Conrad, has written this beautiful introduction to geometric class field theory in his characteristically lucid style.
2) Another friend, Péter Tóth, has just written a Master Thesis "Geometric Abelian Class Field Theory" which seems to be what you are looking for: it is geometric and contains all necessary prerequisites. And the author writes in his abstract that he wants "...to remedy the unfortunate situation that the literature on this topic is very deficient, partial and sketchy written...".
3) David Ben-Zvi, a well-known specialist and another friend of ours, gave talks on Geometric Langlands at MSRI in 2002, and part I is on geometric class field theory. Here is a link
I am very happy and proud that all the specialists mentioned above are members of and active contributors to our site.
I have seen Section e) of the letter from Deligne to Serre available at http://www.math.uni-bonn.de/people/richarz/DeligneAnSerreFeb74.pdf mentioned as a reference. I have not read it myself and it is hand written (in French), but it might be what you are looking for.
There is also Milne's Arithmetic Duality Theorems, http://jmilne.org/math/Books/ADTnot.pdf p. 126ff., Appendix I.A. He proves local and global class field theory using algebraic geometry and Galois cohomology.