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How to measure I-V curve for the LED

Ideally, we need two DMM: one to measure the current, and one to measure the voltage simultaneously. However, you have only one, hence we have to make do with the below.

In the configuration below, the measured voltage is directly across the LED, but the current probe polarity is reverse of that of the LED and you will read a negative value. Just flip the sign. This is how you vary the current:
1- push the DC I button.
2- vary the coarse and fine voltage power supply knob to read approximately the current value you want. No need to be precise. For example. you want to set 10 mA, it is OK to get anything from 7 mA to 13 mA.
3- push the DC V button and read the voltage. If the Auto scale is not on, just push its button. 

It is recommended that you use hook connectors and avoid hooking the connector onto the LED leg. Use a short wire with one end plugged into a hole and the other end for the connector. That way, you don't touch the LED throughout the measurement process.

This is an example what you should record in your lab notebook:

current (mA)	voltage (V)
0.	0.5
0.	1.5
4.211	1.653
12.75	1.771
19.3	and so on...

and so on...   do it for current from 0 to ~30 - 35 mA. You can go to 50 mA for red, orange, yellow if you wish. Do not go much more above 30 mA for the 480-nm blue LED. If you really want stable and low-error measurements, you can short the two leads of current measurement together each time you measure the voltage. Below is an example of a red LED distributed to the class. The points are data and the curves are model fit.

If you wish to see both positive sign for voltage and current, use the configuration below.

In the configuration below, the voltage is measured across the LED and the current probe of which the impedance is negligible and hence its voltage is also negligible. In this case, both the voltage and current will have positive sign. Read the steps above to set the current, read voltage and record in your lab notebook.