I agree. You need constant current if the voltage is going to change, such as battery voltage dropping over time. But here you have 5V. If that's a regulated supply, then you just need the right resistor for the Vf of the LED.
And have you said what the Vf of the LEDs is at the desired current?
Edit: It's times like this that expose my lack of an EE degree. I guess constant current is really to deal with variations in Vf.
Anyway, If the TIP31 has a gain of 25 at 1A, then the base current needs to be 40mA. And if the emitter is at 0.7V, then the base will be at 1.4V. So you have a voltage drop of 1.9V across the base resistor, which would need to be about 47R. That's assuming your MCU GPIO will source 40mA. Seems like a mosfet would be a better choice, and if possible, two LEDs in series.