So I have all the components from the welder I just have to build a case for it. Once I have that done then I'm thinking of bypassing the control board and just using an arduino. I do need a few components, digital readouts and a pot for the wire speed. This is possible right?
You'll need something to interface from the Arduino to actually switch the motor power (MOSFET h-bridge for instance) and some sensors to give you something to display.
If you're monitoring output voltage then you will need some galvanic isolation, and depending if you want to know the voltage at the input to the rectifier, output of the rectifier or at the output of the unit the nature of the voltage sensing will be slightly different.
Current can be easily enough done with a current transformer at the output of the transformer (measuring the AC).
As prior warning, it won't be the easiest project because there will be a lot of EMI that will easily couple into your circuits if they're not designed well.