You mean building a controllable lamp fixture ?
The first consideration is it is mains powered and, given the limited space you'd have for any circuitry, the prefered option would be to directly power the PICAXE from a direct to mains connection ... and if you think people here asking about connecting to cars get short shrift it's a hundred times worse when it comes to direct mains connection
Working on mains is inherently risky and you need a safe working and debugging environment and I consider it essential that someone else is always there to keep an eye on things in case the worst happens and that person has to be trained in resuscitation techniques; we don't want to see your magic smokes drift out the window
X10 uses a high-speed signalling system with data super-imposed on the line. The PICAXE is not able to deal with that signal itself, it's too slow.You could impose serial data on the mains but you'll soon find why X10 does it the way it does and you'd have to design not just the lamp switcher but also the plug-in controller elsewhere. It would also not be compatible with X10.
I tried X10 ( or whatever its UK equivalent is ) and I wasn't that impressed. Signalling did not seem as reliable as I'd hoped, it was expensive, and, to be honest, I could not find any practical application for it. To switch a light on or off I'd probably go for relay, SSR or triac switching the mains power.