How does the voltage doubler work at the startup?
The doubler bootstraps its bias voltages across the capacitors until it reaches a steady-state where the output is double the half-wave amplitude of the AC input.

You can visualize the doubler's behavior using a simulation, right here using the CircuitLab tool.

Here's an example:







simulate this circuit &ndash; Schematic created using CircuitLab

And here's a quadrupler, showing how the stages stack up:







simulate this circuit

A detail you need to consider is that the diodes have Vf drop, which will reduce the voltage the doubler can reach. I've shown Schottky types (Vf = 0.3v) to minimize this effect.