You seem to have the basics of a feedback control system, and that looks OK, but some details are missing.
The battery and the Arduino would need a ground connections.
The circuit diagram uses lines to represent wires, and also uses other lines to represent data connections that I would expect to be multiple wires. The form of the temperature measurement isn't clear. I assume the PC is a Personal Computer, and I don't know why there would be any need to have an A/D converter between a PC and an Arduino.
Depending on the form of the wire, how long and thin it is, the current needed to heat it up may be a lot. You need to make sure that the battery can deliver that current and the MOSFET is rated to take that current. It may be necessary to use some sort of transformer or buck regulator to create more current at a lower voltage, and that is another significant engineering challenge.
The basic idea, that the Arduino can read the temperature and then control the current via the MOSFET is fine. You might not need a gate driver if you use a logic-level MOSFET and you aren't switching the current on and off quickly.
If you are getting one of these working, it's important to make sure that each part of the control system works before letting it try to control on its own. You should make sure that the Arduino can read the temperature in conditions where nothing will be damaged if it doesn't, and also you should make sure that it can control the current with a load that won't be damaged if it is left on.