Paul:
Unfortunately, the easiest way would be to purchase a more supported weather station, like a Davis station. While sainlogic appears to have "an app" for sending data to Wunderground I'm not currently aware of any CWOP/APRS software that handles that particular station, but I've not done exhaustive research. CWOP, having it's roots in the Ham radio community, utilizes the APRS protocol to transmit weather data using the APRS message format for weather data. So, you must have a software, either built in or external, which takes the raw weather data from the station and output's it in the APRS message format. If there is no commercial/open source option available for the station, and the vendor does not support it intrinsically, then your only other option would be to write your own. You may also find other
ws0310
owners who have already written their own... but haven't necessarily advertised it or made it widely available.
Once you have the data in APRS format you can start to think about whether you plan to send the data entirely over the terrestrial Internet to one of the public APRS-IS servers,
over the terrestrial Internet to one of the private APRS-IS servers, over RF, or some mixture of the latter two.