What Modulation does weather radio use?
If I remember right, narrow band FM is used.  "Narrow" refers to the small maximum frequency deviation from the carrier center frequency.  In FM, more deviation means more signal to noise ratio.  Less fidelity is required to send simple digitally encoded signals and voice weather reports than good quality music as commercial FM does.  In this case, the deviation is 5 kHz (again, if I remember right).

As for how to build a receiver, that's no different than any other FM receiver.  There are various topologies for that.  At 160 MHz or so, you have to wake up and design such things carefully.  It's not like 1 MHz AM where simple hacks can get reasonable results.