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.
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.