How do I set up a circuit in order to help me measure the resistance of a 1-meter length copper wire with changes in temperature?
Even after ignoring safety concerns, I don't think that heating the resistor by applying larger voltages is the best method. I would prefer a 4-wire (kelvin) measurement at some constant current to produce a couple hundred millivolts at most (I wouldn't want to push more than 100mA or so). For changing temperature I'd want a dedicated oven with temperature control.
There are simpler methods but I would suggest learning a bit more background and seeing if you can do a more suitable experiment.