Determine Attributes of an LED
Like for any other diode forward voltage varies slightly with current:  



You could say it's a constant voltage plus the voltage drop over a resistor.  

\$V_F\$ is usually specified at the nominal current, which is often 20mA. So place a 100\$\Omega\$ series resistor, and vary the voltage of your bench supply (you've gotta have a bench supply!) until the voltage over the resistor is 2V. The voltage over the LED is \$V_F\$.   

You can't measure the maximum allowed current. Higher currents will decrease the LED's lifetime, but there's no exact point where you can say that it breaks.
That said, it's no use to go too high. Most common LEDs will not show much increased brightness above 20mA.