In order to build a radio controlled clock with the arduino, you need a DCF reciever. In this post the receiver board from conrad is being used (Nr. 641138). For the display the Seven-Segment-LED-Backpack from Adafruit is utilized. - You can of course replace it with any display you like (an LCD for example). The circuit:
|DCF + Arduino + Seven-Segment-Display|
The pullup-resistor and the filtercap are necessary for the reciever board to work. They aren't needed for the 7-seg-display.
|Circuit on breadboard|
In order for the DCF-code to work you'll need to add three Arduino-libraries to the IDE (Sketch->Include Library-> Add .ZIP Library): DCF77, Time and Timezone.
I had trouble with compiling the examples included in the DCF77-library. Updating the Time-library resolved that problem (link).
In order to run the 7-seg-display you'll need the library from adafruit: github-link
Above code is set for the central european timezone. If you live somewhere else, in the UK for example, change the value of "LocalTime" in the fifth to last line to "UK.toLocal(DCFtime)". The same goes for other timezones (they get declared / are listed at the beginning of the code).
It takes around two minutes for the clock to get the current time. You can follow the process by opening the Serial Monitor of the Arduino IDE.