Walta's diagram is accurate, I drew the same thing but did use a circular lamp.
Power goes through the choke and goes to one end of the lamp. Neutral goed to the opposite end of the lamp. The starter connects to the second pin on each end of the lamp. When the lamp is cold the starter closes the circuit and heats the filaments. After a few seconds the starter opens and this also provides an inductive kick that creates a higher voltage to get the arc started in the lamp.
A circline is slightly confusing because the ends of the bulb are right next to each other and hidden in a plastic cap.
These circuits are pretty simple and extremely reliable. I see no reason to cobble up the clock by trying to convert to a different light source.