I have an input signal from a function generator of 1kHz. At 0V the LED should not light up. At 0.5V the LED is partially on, and at 1 V the LED is fully on. I'm not sure what direction I should be going in. Can someone provide me with a holistic approach?

 

Are you saying that your unsure as to how to connect the LED? That would only be a unequivocal concern when powering the LED with DC. Since you are powering it via AC, either polarity is acceptable.

Austin

 

I'm not sure how to power the LED from AC. Is it just like a diode and the appropriate resistor?

 

An LED needs 1.8V for a red one to 3.5V for a blue or white one. They need current that your signal generator might not be able to provide.
So you need an amplifier to increase the 0.5V to 1V signal and provide enough current for an LED.