The most efficient way to drive a MOSFET ( Even a Logic Level type) is via a dedicated MOSFET driver. Microchip has quite a few different models that will do the job quite well. However this advise is many times ignored, especially by those inexperienced with driving FETs.
Microchip MOSFET Drivers
Why a driver? Because a Picaxe can only source about 20 ma of current. While MOSFETs are voltage controlled devices, it takes time for the current to overcome the gate capacitance and the gate voltage to rise. The faster the gate capacitance can be overcome, the less time the FET stays in the linear region and the less heat is generated. This is particularly true at higher switching speeds. If you must drive the FET directly from a Picaxe I/O pin, actual testing has shown that peak current is limited to ~20ma with a 270 ohm series resistor and a 5V supply. This is ok for switching at speeds below ~1000Hz but CERTAINLY NOT for 64Khz.
LEDs do not need to be driven at high frequencies. Typical PWM frequencies are from around 100Hz to 1Khz. With a Picaxe the frequency should be selected so that the duty cycle has good granularity. For example, with a Picaxe 20M2 operating at 16Mhz, a PWM Frequency of 100Hz is not possible. The lowest possible frequency is 245Hz.
A frequency of 250Hz will give a resolution of .1 percent duty. (1000 steps). This may be a good choice.