Controlling a light bulb connected to the mains requires a SSR, Solid State Relay. There are several kinds, the type you want is known as a phase angle dimmer, or proportional dimmer. You feed the input with signals from the PWM output to change the brightness in response to the changing brightness of your flame.
However the fact that you are asking the question, and the manor you asked it, would change my advice to please ask some one who knows about electricity to wire this up for you. Mains can and does kill and I would hate for you to be eligible for a Darwin Award.