With the full-frame camera, should I expect to see the entire circular image?
It depends on what focal length you've set the lens to. The EF 8-15mm f/4L USM fisheye is unique in being circular at the wide end (at least on full frame it is), and diagonal at the "tele" end. Where you are in the zoom range determines how much of the frame is covered by the lens's image circle.
Most fisheyes are prime lenses, and are either circular (fit the entire image circle in the frame) or diagonal (fit the entire frame inside the image circle). The typical focal length for a full frame circular is 8mm, and for a diagonal is 15mm. On APS-C, the only circular I know of is the Sigma 4.5mm, and most APS-C 8mm lenses are diagonal.
If not, what do I need to do in order to get the entire circular image in frame?
You need to have the 8-15L's zoom set to 8mm.
See also: the 8-15L review on the-digital-picture.com, it has a graphic example, of what the lens will look like on full frame at different focal length settings, where you can mouse over the zoom setting, and see the resultant frame.