Like most diodes, LEDs are constant voltage devices. That means the voltage across them remains fairly steady as the current through them changes. Their brightness is related to the current, not the voltage and it will increase with the current.
In your photograph, I presume (blinded by the light!) there is already some series resistor in the connection or the LED would most likely burn out when connected across the battery. Ideally, you need a variable power supply with a current limit circuit. You set the current limit to say 10mA for a small LED like yours up say 500mA for a high power LED and slowly increase the voltage. When the current reaches 10mA, you read the voltage and that is the Vf you are looking for.
If you do not have an adjustable power supply, make an initial guess at the voltage, it would be somewhere around 3V and find a power source a few volts higher, maybe a 6V or 12V battery. Work out the series resistor to limit the current using this formula:
(source voltage - 3)/desired current in Amps
So if you used 12V and 10mA the resistor would be 9/0.01 = 900 Ohms. Connect the LED and resistor across the 12V so it lights up. Then using a testmeter, measure the voltage across the LED itself, that is the Vf.
Note that LEDs like the ones you show are rarely rated for more than about 20mA.
Brian.