I can’t see anything about an alarm function in the DS1307 datasheet. The square wave output outputs a square wave (!) at a minimum frequency of 1 Hz, so that doesn’t help you much - you will need a microcontroller.
Microcontrollers don’t get much simpler than the PICAXE range, the 08M2 chip would do fine. For minimum hassle I would get something like this starter pack which includes the programming cable and a prototyping board, but once you’ve programmed the chip you can use it on its own - just tie its Serial In pin to ground.
The DS1307 is used in one of the PICAXE add-on boards so you should be able to get plenty of help on the PICAXE forum if you need it.