<p>How do I create an ASP.NET web service that returns JSON formatted data?</p>

The most important thing to understand is to know how to represent data in JSON format.
Please refer
http://www.json.org/
to know more about it.
Once you understand this, then the rest part is pretty straight forward.
Please check the following URL for an example of the same.
http://www.ajaxprojects.com/ajax/tutorialdetails.php?itemid=264
http://code.msdn.microsoft.com/JSONSampleDotNet
http://www.phdcc.com/xml2json.htm
I recommend Jquery library for this.  It's a lightweight rich library which supports calling web services, handle json data format output etc.
Refer www.jquery.com for more info.