If your signal source is something sharper already (like a square wave, but maybe not very good risetime), you can skip all the biasing junk and use a voltage divider from input to base to GND. You design the voltage divider so that it has a base voltage of about 1.4V (assuming the base didn't act as a diode), and a Thevenin resistance of enough to deliver the required base current (usually, Ib = Ic / 20 or so). This way, when the source is 'on', the base gets +Ib, holding the transistor in saturation (a 2N3904 will turn on in ~30ns, depending on various conditions). When the source turns off, the base capacitance is discharged to GND, with about -Ib of current (a 2N3904 will probably turn off in 100ns under these conditions).
As I mentioned earlier, you can cascade stages to increase the gain and reduce the risetime, at least to a point. A "switch" circuit like this is really just a linear amplifier, at some point of its swing -- it's just not intended to stay in that range very long, hence the lack of biasing! After a few stages, if the risetime is now limited by the circuit itself (and not the signal source), you need to change up the circuit. Usually, more -Ib, more Ic, and lower resistances is required to get there. Under ideal conditions, you can get ~10ns edges out of a 2N3904.
Tim