Cannot alter output of a view under any circumstances
When you use Custom Text, you are telling Views to override anything it does and display only what you have provided it plus the addition of your placeholders. So what you should do here is exclude the original field from the display and write your HTML with the class into the custom text field. Then, include the field placeholder.

Like this.

  &lt;a class="my-class" href="[field_link_url]"&gt;[field_link_title]&lt;/a&gt;


I am pretty sure this gets you the intended result. Note I am making up those placeholders and you should see what your available replacement patterns are. :)