It's an interesting idea, and one I've thought of doing a few times. I have a small (A4 size) CNC desktop router that I bought from China a fair time ago with a view to using it for PCB making. I've never really got to grips with routing out PCBs with it, for several reasons, one being that the PCB has to be absolutely dead flat for it to work, and I never got around to making something like a semi-disposable vacuum table to hold the boards in position (double sided tape on a bit of thin scrap ply didn't seem even enough for track milling).
I don't have any problems at all etching PCBs, but even with the accurate, high speed, bench drill I have (a proper PCB drill, with a 3mm chuck) drilling is tedious.
In essence, the control bit is easy. Start with a known reference point (put the board on the bed and then manually position the drill very accurately at the reference corner/hole) and then move to each X-Y coordinate in turn, lower the Z axis to drill, raise it, then move to the next coordinate and repeat. The Picaxe bit is pretty easy, just store the X-Y coordinates of each point, then it's a pretty simple bit of code to move to each in turn. For hobby level boards, that aren't large or have lots of holes, this should be fairly straightforward. The interface is easy, too, just use some cheap stepper drivers and then all the Picaxe needs to do is keep track of X and Y and execute a bit of code to send the right pulse and direction outputs to those axes, together with a bit of code to lift and lower the Z axis (the drill).
The challenge is probably in getting a meaningful set of X-Y coordinates from Diptrace. I've no idea how coordinates are stored in these files, but if there is a way to get them out as text, then it should be pretty straightforward to either pass them to a Picaxe or even just load them in to it via the programming lead.
The whole thing needs a fair bit more thought; the above has just been my initial thoughts, but I think it should be well within the bounds of possibility for such a fairly simple task.
My experience has been that the learning curve for CNC machines is steep, which is why I have a new and almost unused desktop router sat here and a load of parts to build a very much larger router. It seems that the majority of these things are aimed at doing very complex stuff and have software that does everything imaginable. The snag is, if all you want to do is drill a pattern of holes very accurately the thing is far too complex to get to grips with. I'll admit to not having looked again recently, as it's three years or more since I last played with it, so perhaps there are simpler options now available.