You need to install -dev package for zlib1g - it is named zlib1g-dev:
sudo apt-get install zlib1g-dev
and it will install zlib.pc file for pkg-config along with zlib.h header.
For pip to work, Python needs to be linked to the zlib library when Python itself is installed. It appears that either zlib was not installed when you installed Python, or at least that the Python installer couldn't locate it. To help it along, you may issue the following before installing Python. In bash syntax,
Copyzlib_lib="/usr/lib32"
zlib_inc="/usr/include"
export CPPFLAGS="-I${zlib_inc} ${CPPFLAGS}"
export LD_LIBRARY_PATH="${zlib_lib}:${LD_LIBRARY_PATH}"
export LDFLAGS="-L${zlib_lib} -Wl,-rpath=${zlib_lib} ${LDFLAGS}"
Here I've assumed that zlib is installed under /usr/lib32 and /usr/include/. To check this, look for the libz.so.1 file in the "lib" directory and the zlib.h file in the "inc" directory. If you find them somewhere else, simply change zlib_lib and zlib_inc accordingly.
IF you have different versions of Python installed it is likely the install is on another version. for example. I have pyperclip in 3.6.3 32bit version, but I an not access it in 3.6.3 64bit or 3.7.2dev.
Ubuntu 16.10+ and Python 3.7 dev
sudo apt-get install zlib1g-dev
Note: I only put this here because it was the top search result for the error, but this resolved my issue.
Update: also the case for ubuntu 14.04LTS and base kernel at 4.1+
The solution is : # yum install zlib-devel