Hi Marcus,
I've heard of using an eeprom in just the way you talked about. The concept you are talking about is called a Look Up Table. While the EEprom will work dealing with the logic to implement the correct timing might be more trouble than it's worth. A microcontroller will let you do this same idea but remove a lot of electronics allowing you do do mostly software- which is a lot easier for a beginner.
Another posibility, since your graph looks like an exponential curve, would be to use a logarithmic amplifier. A logarithmic amplifier is just an op amp with a Diode or bipolar transistor in its feedback loop. Some tuning would be reqired but it might work.
A note about your wording: when you say volts DC that means it doesn't change- like a power supply. What you mean is a signal from 0-5V.
Hope this helps.
Brent