How do I programmatically add and fill a field to taxonomy terms of a specific vocabulary?
This is really two questions. 

For the first, I would use hook_taxonomy_term_view to alter $build['description']['#markup'], like so:

function custom_taxonomy_term_view_alter(&amp;$build) {
  $build['description']['#markup'] .= '&lt;p&gt; And this is more stuff.&lt;p&gt;';
}


For the second, taxonomy_term_save() can take a description field as described on the API docs page.