The only required package for a standard installation is jasper which enables the jpeg2000 packing/unpacking algorithm. It is possible to build grib_api without jasper, by using the --disable-jpeg configure option, but to install a fully functional library, its download is recommended.
> gunzip grib_api-X.X.X.tar.gz > tar xf grib_api-X.X.X.tar
> mkdir grib_api_dir
> cd grib_api-X.X.X > ./configure --prefix=grib_api_dir
> make ... > make check ... > make install ...