How do I calculate the voltage needed to heat up a wire to a certain temperature?
I don't believe you can do this with any degree of precision : it depends too much on the wire and its environment which determines its rate of cooling. 

This 450 page book would give you a brief introduction to the difficulty; the TL/DR version is: Don't.

Which leaves you with the alternative problems of: 


determine the wire's temperature and control the voltage to achieve the correct temperature. You could measure the drive current and voltage, to determine its resistance, and compute the temperature from that
measure the actual motion and control the voltage accordingly (programming a PID controller is the best way, but the response time of the wire makes this tricky)
Experimenting to determine what voltage is "good enough" in your environment. This is by far the simplest and may be good enough for your application; you haven't told us enough to tell.