"Dimming" AC power electronically is quite a tricky task.
Manual dimmers are effectively a large potentiometer (actually called a Rheostat). To do it electronically is somewhat harder.
The accepted method of doing it is kind of like PWM, but with special timing. The AC waveform is chopped off at certain percentages through its cycle to reduce the power. This takes special hardware, and interfacing it is not a task I would enjoy.
Your best bet would be to get an infra-red dimmer plate to replace your existing wall switch, and program the Arduino to send the right remote codes through an infra-red LED. That way you have the remote control as well 