The Amilia API is located here: http://amiliaapp.github.io/amilia-demo-organization/api-documentation.html. It lists calls that are public. In addition, there is another document which lists calls with authentication required. Those are not yet public and will be sent in a separate PDF.
First off, most calls take as argument the language (en for English in your case) and a "rewrite url" - the id of the organization or customer. For your demo org, it is myskillchart. You'll notice those parts in the signature of the API calls below. I built the examples using your demo account.
The API is always a URL and returns JSON. Authenticated calls require an extra HTTP header. I've gone ahead and used jQuery to code up the calls. You can run those in your browser from your website domain or on the amilia.com domain (we respect CORS).