You probably have one the Toyota Accessory wheels like these right?
I've seen 225/60R15s on a gen4 camry before with stock alloys (same size 15 X 6). It will fit.
However, those are pretty wide tires on a narrow width rim (6"). Your sidewalls will bulge out and have more flex which will actually degrade the handling a bit.
If you want to keep the stock wheels, I would look for good performance tires in the same size rather than going wider.
I installed 205/60R16s on my Camry along with aftermarket 16" wheels. The handling sharpened up a bit and it still maintained a good ride.