First of all you need a java runtime environment installed, eg. from sun/oracle.
Unzip the release archive file to a folder on your local media. It can be something like C:\Program Files\EarthdawnCharacterEditor\. Any new release can be unziped into the same folder, but you have to allow your unzip tool to overwrite exiting files. You may have changed some files in the sub folder "config", for example the file optionalrules.xml. You may want to backup those files before installing a new release.
For Linux only, be aware that a zip-archive do not preserve the execute-attribute of files. So you have to execute chmod +x EarthdawnCharacterEditor.sh, if you want to run EarthdawnCharacterEditor.sh without a shell as prefix.
Unzip the addon distribution archive file to the same folder on your local media, where you have already installed the release archive file before. Unzip will ask you to overwrite some files. You need to overwrite the existing files, because the new files contains additional information needed from the addons. Remember to reinstall the addons when you update the main distribution.
You can safely install the addon distribution while the Earthdawn-Character-Editor is running. But the changes will first become effectively after restarting the Earthdawn-Character-Editor.