You've got an incoming clock signal. If you can use that to switch the transistor in the converter, then it's fairly easy. To start with, you need to know the frequency and duty cycle.
Best case:
- It's a fairly high frequency (say 100KHz), so you can use a small inductor.
- The duty cycle close to 50%, so you automatically get the right output voltage.
You can just choose an inductor value that won't give too much ripple, calculate the minimum load current, and put a resistor across the output that will draw that much current from the negative supply. Remember that with too little current drawn from the output, the output voltage will rise too high.
Harder:
- The duty cycle is wrong, say 25%.
Now you have two choices:
A) Use the given clock signal to generate a new clock with the duty cycle you want.
B) Make a converter that wants to create a higher output voltage than you want, and run it in burst mode. i.e. Monitor the output voltage, switch off the converter when the voltage goes above 5.5V, and switch it on again when the output voltage drops below 4.5V. This is messy because the output voltage is never steady, it's always rising and falling.
Also hard:
- The frequency is too low, say 1KHz.
That would require a very large inductor, which isn't very practical, so it would be better to make your own oscillator and use that to drive the converter.
One more thing:
When you switch the converter on, the output voltage may tend to rise too high before settling down to the correct value. The easiest way to deal with that is probably just to put a 5.6V zener across the output.