There are several issues.
1) You need to install bzip2 libs. Download zipped libs from Bzip2 for Windows, Binaries. Unpack them to some directory, say C:\GnuWin32\bzip2-1.0.5-bin. lib subdir should contain libbz2.a, libbz2.def, libbz2.dll.a. This path (C:\GnuWin32\bzip2-1.0.5-bin\lib) should be provided to --with-bz2-lib.
2) The package bzip2-ruby seems to have a not-fixed error yet according to the author github.com/brianmario (version.h is absent). So you can install bzip2-ruby-rb20 as he suggests.
3) Now install bzip2-ruby-rb20 providing include and library dirs for bzip2 (installed in 1):
gem install bzip2-ruby-rb20 -- --with-bz2-include="C:\GnuWin32\bzip2-1.0.5-bin\include" -- --with-bz2-lib="C:\GnuWin32\bzip2-1.0.5-bin\lib"
Should be done: "1 gem installed"