Use relative paths in your embedding code and the gallery should display both locally (in supported browsers - please see this FAQ for details) and online.

For example:

<!--START JUICEBOX EMBED-->
<script src="../../Benutzer/ThomasKlinger/Bilder/galerie/jbcore/juicebox.js"></script>
<script>
    new juicebox({
        baseUrl : '../../Benutzer/ThomasKlinger/Bilder/galerie/',
        containerId : 'juicebox-container',
        galleryWidth : '800',
        galleryHeight : '600',
        backgroundColor: '#222222'
    });
</script>
<div id="juicebox-container"></div>
<!--END JUICEBOX EMBED-->

The above is just an example. I do not know the file structure of your site so you will need to make sure that the relative paths are correct.

Just for the record, if you added 'file:///' to the beginning of your paths, your gallery might work locally but it would not work online as visitors to your web site would not have access to files on your C:/ drive.
For example:

<script src="file:///C:/Benutzer/ThomasKlinger/Bilder/galerie/jbcore/juicebox.js"></script>