promote.php

<?php 
/**say the students are pulled from a database table via a while loop (or what you're using)
it may look like this:
*/
?>
  <?php while ($row = $yourTable->fetchObject()): ?>
  <tr>
    <td>$row->studentName</td>
    the rest of the pull
    ...
  </tr>


Now you can add a new table data to hold the button, a link preferably:
  <td><a href="" class="someBtnClass">Promote</a>

you'll need to pass the student by their id via a get request 

  <td><a href="yourscript.php?student=<?= $row->id;?>"> promote </a>
  note you're still in the loop so you can access the

since the id is being grabbed, the next thing is to run the query 
<?php 
// this code will be above the loop (best practice)

if (isset($_GET['student'])) {
  $studentId = (int)$_GET['student']; // you can run extra checks here for empty data 

  // query 
  $promoteQuery = $dbDriver->query("UPDATE students SET class = '$theDataForTheClass' WHERE id = '$studentId'");
}