Tor Sterner,
the thing is that you have to provide the search string to Google Analytics in URL as a query parameter (in your example, it's named q). In GA web interface, you can then set up the site search settings, fill out the search parameter and that's it.
You were for that reason sending virtual pageviews, and that's what you need to do with GTM as well. However, the better approach would be to simply append the search parameter to URL, something like:
By default, GTM uses url path for Document Path (=Page) dimension. If you append your search string, then yon can setup search query parametr in Analytics interface (View Settings > Site Search) and that's it.
I have used {{Search String}} macro to capture the search phrase - but since I am not sure how search is implemented on your site, you will have to come up with a way how to do this (DataLayer variable, Custom Javascript macro etc.) - and make sure it returns the search query with parameter as well. It should look like this:
?q=searching