BJT based mosfet driver, what resistor values to use?
24V is probably above the max Vgs for your FET so it will smoke. You need to limit Vgs to a safe value. For example you can add a resistor from Q2 base to ground to form a voltage divider with R2.
When Q1 turns on, Q3 base current is not limited, and it will exceed the maximum ratings for BC857. Q3 would need a base resistor.
There is also the issue of hFe and bipolar transistor current: when swinging the FET gate towards Vcc, Q2's base voltage comes from the 10k resistor which is quite a high value. Its base current will be quite small, and BC847 can only output a couple tens of mA before its hFe falls off. So the output current you'll get from this driver probably won't be more than a hundred mA. This should be enough for low frequency PWM though.