There are a few different ways of filtering the output in GeoServer, though none of them are "OGC standard" though I doubt that matters in this case. The one that is relatively simple to get to grips with is CQL, described at http://docs.geoserver.org/latest/en/user/tutorials/cql/cql_tutorial.html. You simply add a parameter called CQL_FILTER to the URL, setting it to the filter you need (URL encoded of course, e.g. spaces should be represented as %20, equals as %3D). So the first thing you need is a filter on your website ID, e.g. the following request gets website ID 1's records:
testwarehouse.indicia.org.uk:8080/geoserver/indicia/wms/kml?layers=indicia:cache_occurrences&CQL_FILTER=website_id%3D1

Note that I recommend you switch to the cache_occurrences layer as it is faster and has less load on the server.

If you still want a custom view added, we need to know which fields are going to be required for each point on the map (i.e. if they click on the point in Google Earth, what info is available), plus what filter criteria you want applied.