How to bias a class C amplifier with negative voltage?
There are many options. A simple one is a voltage inverter like this:



Here the negative voltage is unregulated.
Note that the circuit around the 555 only generates a square wave.
If a suitable clock is already available in a product then this could be used as a clock source as well. Depending on how much current is needed from the negative rail a buffer might be needed to charge/discharge C2 more quickly.

There are also special ICs for this which include a feedback to stabilize the negative voltage.