It depends on the voltage of the camera and what voltage the appliance runs on. If it's mains voltage all the way into the clock then you need to seek out the transformer and rectifying circuit to find the DC voltage see..
Mains supply from appliances for spycams.
Those mains powered devices are getting rarer, chances are you can take a supply direct from the appliance adapter usually 5v-12v DC (if your camera will operate at those voltages.).. but connect to the DC voltage inside the appliance and not at the adapter as EB suggested.. that way you avoid a give away "extra cable". although you still will need to get your signal cable out unless you are recording in the appliance itself.
Wired cameras tend to come in three voltages 3v-5v DC, 6v-9v and 9v-12v, hopefully you will get one to match your clock adapter other wise do as EB said and take 120v into the clock, strip down your camera AC-DC adapter and fit that into the clock.. if you've got enough space in there otherwise it's back to the drawing board!!