This is the basic circuit you're looking for:
(Or one of the special-purpose all-in-one ICs that does the same basic thing.)
The average circuit you find on the 'net has all the resistance in Rb and no Rbias and Cbb. A gentle reminder that, just because a thousand parrots squawk the same tune, doesn't mean it's "right"...
Best efficiency is usually found with zero Rb, Rbias sufficient to get the current flow you need, and Cbb sized so the Rbias * Cbb time constant is around the inductor time constant.
What's the inductor time constant? Well, here's how it works.
As Cbb charges, the transistor begins to turn on, then positive feedback through the transformer makes it slam on.
Switched on, current charges up in the inductor (the fundamental inductor equation is V = L * dI/dt, so when we apply a fixed voltage V, current rises by dI every dt seconds). Meanwhile, base current is initially large, but tapering off (because the base winding is drawing current from Cbb, discharging it). Eventually, there's not enough base current, and too much collector current, and the voltage swings up. At that instant, base voltage drops sharply, turning the transistor off.
After turn-off, the collector voltage swings up until it hits something. In this diagram, you'd probably see a spike of 30V or more, and a very hot transistor.
Normally, you put an LED from collector to GND, so the voltage jumps up to a well defined ~3.6V and the current discharges into the LED, lighting it.
As mentioned, it would be difficult to measure the voltage and current in an LED driver. We can address this by adding extra steps:
On the left, Rbias has been replaced with a PNP transistor (this, and the other PNP transistor, are optional for demonstration purposes). Recognize Cbb, the NPN, transformer and Rb are in the usual connection. A schottky diode takes the inductor's discharge current and directs it into a 10uF capacitor, which smooths the pulses into DC. (the 1k and Vref diode are also part of the control in this circuit, and are optional.)
To measure the DC performance of this circuit, measure the voltage and current at Vin (note: you need a 10uF capacitor from Vin to GND as well). Measure the voltage and current out of Vout, when loaded with a suitable value resistor, or some LEDs.
The schottky diode costs a little efficiency, but not too badly: a 20V low-loss schottky (I'd recommend a PMEGxxxx) will drop less than 0.5V, not a bad price for getting a stable reading.
Tim