Yes, I am trying to build an e-cig.
That's what I thought.
Measuring and/or control
You don't have to worry about feeding a specific voltage, power or current to the coil. All you need is the ability to decrease or increase the "oomph". You can do that by varying the PWM duty cycle.
If you need to know the output voltage, current or power, you can measure the output. If you want to be able to set exact values, you can take this value and make a control loop.
But even measuring the output can get complicated real fast, not to speak of making a control loop and a user interface for it. So if I were you, I would leave the measuring and/or controlling part out for now. You can always add it to a later project.
A simple potentiometer connected to an analog input of the Arduino can serve as a dumb "volume button" for your mod.
MOSFET gates and Arduino outputs
The outputs on the Arduino are too weak to drive a normal powerMOSFET directly with PWM. They're strong enough to turn a logic-level MOSFET on and off slowly (for instance when the user presses a button), but if you want to drive the MOSFET at PWM frequencies, the MOSFET will require a higher gate current than the Arduino can provide.
Solution: Choose a suitable MOSFET driver, or design your own, and place it between the Arduino and the MOSFET.
It is probably a good idea to take some time to read more about MOSFETs and MOSFET drivers in general, before you choose which MOSFET you want to use, and how you want to drive it.
For an e-cig you probably don't need to smooth the output power in any way.
"Rattlesnaking" is not an issue with modern low resistance builds because of the high thermal mass of the coil. Even with thin wire, a PWM frequency of more than a couple hundred Hz will practically eliminate anything noticable.
A flyback diode won't hurt, but I don't know if it will be necessary. The inductance of an e-cig coil is probably negligible.