How would one go about using a 12 V DC power source to power something which needs 4.5 V DC using resistors? Is there a way to determine how much adding a resistor would drop the voltage?
The short answer is "don't do that." 

The voltage dropped by a resistor is given by Ohm's Law: V = I R.

So if you know exactly how much current your device will draw, you could choose a resistor to drop exactly 7.5 V, and leave 4.5 V for your device, when that current is run through it. But if the current through your device is changing, or if you want to make more than one system and not every device is exactly alike in current draw, you can't consistently get 4.5 V at the device using just a resistor.

Your other options include


A linear regulator. This is basically a variable resistor that will adjust it's value to keep the output where you want it. This is probably only a good solution if your device draws very little power (maybe up to 100 mA).
A shunt regulator. This means using a resistor to drop the voltage like you are suggesting, but then adding an extra device in parallel with the load to control the voltage. The shunt regulator will adjust its current (within limits) to keep the current through the resistor correct to maintain the desired output voltage.
A switching regulator. This uses some tricks to generate your desired output voltage with much better power efficiency than a linear regulator. This is probably the best choice if your device needs more than 10 or 20 mA of current.

