Control NES via Arduino
If your ATMega is running off its own 5V rail, connecting the grounds should be all you need to do. If the ATMega is running off a different supply voltage (e.g. 3.3V) you would need some level shifting circuitry to avoid damaging the ATMega with the higher level signals presented by the SNES hardware.

(I wonder why the 5V from the SNES wouldn't be good enough to power your ATMega. I can't imagine it drawing tons of current...)