If the edges are lifting when you're removing the plastic, that is a strong indicator that the glue you used is not of a pedigree suitable for this type of repair. The adhesive is not melting into the rubber and not vulcanizing, but only providing a superficial tack, like the back of a scotch tape.
The edges will stick quite well to the inner tube if the glue is good.
However, even if the adhesive is good, to avoid "tempting fate", what I do is peel the plastic backing from the centre out. Before even gluing the patch on, I score a small X crosshatch cut in the center of the plastic backing with a sharp tool (but gently, without damaging the patch, obviously).
When the patch has set (I give it 12-24 hours, clamping the tube between two wooden blocks with a wood clamp), I then peel the plastic backing from its center cut out toward the edges. That direction reduces the tendency to lift.
Another important aspect is that there must be a decent amount of glue there all the way to the edges of the patch, and at least a millimeter beyond. When I'm applying the patch, I wiggle it around in a circular motion to get the glue spread to a slightly larger area than the patch, but still mostly within the limits of the plastic backing sheet (to avoid adhesion to the clamping blocks, and an overall tidy job).
At "peel time", you notice a big difference between the behavior of cheap rubber cement and proper tube repair contact cement: with the good adhesive, the extra glue squeezed out of the patch, caught between the plastic backing and the tube, is almost completely tack-free and releases the plastic backing easily. The poor type of adhesive remains more of a sticky, gooey mess, indicating a product that is for temporary paste-up jobs that allow repositioning. If your glue is like that, of course the edges lift, because in fact the whole patch will lift easily.