From the datasheet:
IC = 3.0 A; IB = 12 mA VCEsat* max. 2.0 V
It also says that the HFE (current gain) is about 1000. As you can see from above, they are overdriving it by about 4x. This is typical for a darlington pair transistor. The rather large VCEsat voltage is typical of a darlington pair, too, but may be more on the order of 1.5V at the 750mA your LEDs draw.
So using the datasheet's preferred ratio of 1:250 base to collector current, 750mA/250 = 3mA.
About 5V will be provided by the Arduino (actually a slight bit less as their is internal resistance) and the VBE drop must be taken into account when calculating the base resistor. VBE at 3A is rated at 2.5V. So about 2.5V will be dropped across the Arduino to Base resistor.
2.5V/3mA = 667 ohms
For the sake of taking care of the slightly less than 5V out of the Arduino, variations amongst transistors and resistors, let's use 470 ohms.
2.5V/470 ohms = 5.3mA well under the maximum base current.
Say worst case, the Arduino manages to put out only 4.5V because of a depressed Vcc, leaving only 2V across the resistor:
2V/470 ohms = 4.3mA well over the 3mA required to drive the TIP120 into saturation.
As for heat dissipation in the transistor, we'll use the 3A rating of VCEsat of 2V, as we don't know exactly how much lower it would be at 750mA because it isn't linear. And we should err on the side of caution, or risk burning up the transistor:
750mA x 2V = 1.5W
A TO220 package will burn up with more than 1W, and even at 1W you are taking a risk. You should also double that number to take into account changes in ambient temperature, objects near the heat sink that interfere with air flow, dust, etc.
But 3W is still not bad. Simply bolting it to an aluminum case with the proper insulation and heat sink paste (more is NOT better) or to a reasonably sized heat sink should suffice. If you use a silpad, do NOT be tempted to add heat sink paste.