If your goal is to remove the PC (a good goal), you would need a way to get the data from the weather station (transmitting on either 433 or 900MHz presumably) to a computer somewhere. If the weather station's base has a USB then it might work on a RPi, but that depends on if someone has written an interface. If not, you would have to figure something out. I'd recommend searching for your station and or the brand and Linux/RPi to see if it's already been done.
If your goal is to keep the station plugged into the PC and use that to transmit, that's pretty easy. You simply set up a MQTT client on the PC and use a script to send the data to a MQTT broker. I think some of the weather trackers (WU perhaps) currently still have an MQTT broker to upload data to...maybe not. Either way, you can send it directly to HG once a broker is set up on the RPi. You will need something inside HG to catch the data. Look at my MQTT code to see how I did this for some temperature sensors and weather data. Others have probably done something similar if you look around online.