This line
#include <hdf5.h>
does not refer to the file ifself but to another file with the same name expected to be located typically under /usr/include
or whatever your compiler's default path for include files is.
However you seem to be missing the hdf5 library developer files, which should include /usr/include/hdf5.h
You might either find them here: http://www.hdfgroup.org/downloads/index.html or in the repository of your OS distribution.