If you have the recordset working then you should be able to just add in a URL parameter that the client detail page is looking for. When you do a search for clients what does the URL look like when you go the details of a particular client?
You will need to replicate the same URL parameter and value when linking to the client details page. It should have the client id in the URL parameter, you could use the client id value from your recordset for the value of this URL parameter.
The link will look something like this:
<a href="client_details.php?clientID=<?php echo $row_YourRS['clientID']; ?>">link to client details</a>