The National Oceanic and Atmospheric Administration is responsible for broadcasting the signals used in weather radios. They use a protocol called Specific Area Message Encoding (SAME) and [Ray Dees] recently published an Arduino library that lets you decode the SAME message packets.
He doesn’t provide a method of tuning the radio signal, but at first you can use the audio samples he points to. The actual broadcasts happen on one of seven frequencies between 162.400 MHz and 162.550 MHz but the tones are also broadcast on TV and Radio alerts. Once you have the audio it is fed into a pair of XR-2211 Tone decoders. This provides just three interface pins for the Arduino to watch.
The annoying noise that grabs your attention at the beginning of a weather alert, or test of the alert system is actually what the SAME data packets sound like. From those tones this system will be able to decode what type of alert is being issued, and the geographic locations it affects. If you interested in more info about SAME head over to the Wikipedia article on the topic.