While some PICAXE's can be connected to PS/2 keyboards, it is unlikely that they could parasitically connect in parallel with a keyboard to PC cable; there would be problems with both the PC and PICAXE trying to drive the keyboard clock, which could damage the PICAXE, and more expensively, the PC motherboard itself.
Because of the high speed nature of the PS/2 keyboard interface, it is not possible to put a PICAXE between a keyboard and the PC and do it that way.
The easiest way to store large numbers of keystrokes would be to use I2C EEPROM; the X range of PICAXE's support I2C interfacing.
If you were considering 'snooping', bear in mind the legal aspects of this. In the UK, the Computer Misuse Act 1990 makes it an offense carrying up to six months imprisonment for attempting to obtain account or password details, and up to a five year term if done with the intent to use that information.
Under CMA 1990 c18 1(1)a, it is an offense to even try to guess a password, even if it is guessed wrong or there's no intent to use it.
Most of the 'jolly japes' I was involved in while studying Computer Science ( when <i>hacking </i> was educational learning ) would now be seerely punishable. By today's standards, I should be serving a sentance longer than most serial killers :-o