After downloading, open a shell and cd to the directory where you have downloaded the installer.
At the prompt, type: sh instantjchem-VERSION-linux.sh
After the installation, add the bin subdirectory of the Instant JChem directory to the PATH to be able to run IJC from any directory.
{primary} Instant JChem uses its own Java (that it is bundled in the installer). If you want to use another Java, see How to change IJC's Java.