How do I properly use "that is" in a sentence?

What you've done is fine. There is a subtle style difference between your version and Sicilian Godmother's suggestion. Your version compliments the readers' intelligence by implying that they knew all along what Bernay's chief goal was, and that you're just reminding them, whilst if you use the "...which is ..." version you imply that the reader probably didn't know that before.