Well there's a few ways you can do this: The easy way, the expensive way, and the smaller, more complicated way.
http://www.mit.edu/~vona/bass/bass.html
^Here's a page about hacking the singing fish plaque. I know it's not an alarm clock, but it works in sort of the same way; a signal is sent to make a sound start. It's built around a sound recorder flash memory chip, and will hold a few seconds' worth of music.
The expensive way: Buy a CD-player alarm clock that plays the CD as the alarm.
The easier way: Rig up
this board to the clock. (I'm guessing connect the "play" button to the trace that starts the alarm cycle?)
EDIT: Whoops! Sorry about the huge URL.
