The Deflate compression method is fully, but concisely specified in [RFC1951]. Fortunately, a few books provide in-depth explanations of how Deflate works, and how it is integrated into the PNG and MNG formats.
Data Compression: The Complete Reference, 4th Edition
If you are interested in any aspect of compression, if you are either a beginner or an expert, if you are mathematically capable or a die-hard programmer (or both...), this is a must-have book! It tackles nearly every aspect of data compression (including Deflate and PNG compression, of course), from the information-theoretic basics, to the state-of-the-art methods of text, image and multimedia compression.
Lossless Compression Handbook
Academic Press, 2002
This is a comprehensive collection of contributed chapters, about various lossless compression methods, targeted at intermediate to advanced readers. The section about PNG and MNG is written by Greg Roelofs.
Compressed Image File Formats: JPEG, PNG, GIF, XBM, BMP
ACM Press/Addison Wesley Longman, 1999
This book is oriented towards programmers who need to understand the most popular image formats.
PNG: The Definitive Guide
O'Reilly and Associates, 1999
Updated (2003) and freely available online at http://www.libpng.org/pub/png/pngbook.html
The first book that discussed about PNG and MNG in detail, continues to be both a motivating guide and an authoritative reference.
The author, Greg Roelofs, is also a co-author of the PNG and MNG specifications.