First you'll have to find a Source for this data - that's the hardest part.
This may either be some site that publishes this data as a simple text file or in a more complex format based on parameters passed to it (Such as location etc).
After you have your source it's fairly easy. XML is a good way to format that data - But since quite a few source's don't publish the info in XML format (or a format Flash can directly read in) - You'll have to do some parsing.
Basically you have to write a proxy script that goes out to the external source and either pass's it some parameters via GET or POST - or just reads the text directly in. As soon as you have that info - you can format/parse it however you want. At this point it is a good idea to format your results into XML - but your not limited to that. You can just as easily pass the info back to Flash in a variable/value format - ie Weather1=Sunny mid 80s.
http://www.flash-db.com/Search/ (read up on proxy scripting).
Here's something you may be able to use with some modification:
http://sourceforge.net/projects/phpweather/Another Option is using an existing Web service that publishes this information for anyone to use (I havn't seen a good service that deals with Weather although).
One of the new trends is to use Soap to access these web services. This is similiar to just sending a GET or POST request to an external website - but has some advantages.
You may find a service that offers weather on this list:
http://www.xmethods.com----------------
One thing you'll find with all of this - is that you need something like PHP or ASP to write the actual proxy script for accessing this data. The reason is that security restriction they placed on the Flash player.
Anyways I'll go into more of this later.
Do you have a source to get the weather from - Find that first then it's easier to give direction from their.