First off, the diamond shape itself can be used as a text frame, if that's what you want. You don't need a second frame for text inside the diamond (and if filling the diamond, I don't know how you'd accomplish that without using a diamond shape for the frame). To wrap around the diamond shape, set the wrap to object shape (third button from the left), not bounding box (second button).
If you want to put text on top of an object that has wrap applied, open the text frame options and check the box in the lower left to Ignore Text Wrap.
Use the diamond as a text frame or create the frame and the text separately and then group them. Apply the text wrap to the group.