Too much of this weather software runs as a foreground application. I highly recommend dedicating a pc to only run weather software. It doesn't have to be a monster, and most likely can be an old computer too slow for today's games, media, and applications. I've used a little Pentium 4 with 1 GB of memory for over a year. No issues, always done the job at hand.
As for connecting to your weather server, has anyone tried using Microsoft Remote Desktop (RDP)? We use RDP 99% of time at work to connect to our servers. Only when there is a problem booting into windows do we use the KVM console.
To make sure you connect to the same session as the console (basically what your monitor sees) you need to include a single switch. Depending on the version of RDP you will need to use either '/console' or '/admin' (/admin is for Windows server 2008, Vista SP1, and maybe W2kx if you upgraded to latest RDP client). Be part of all, It doesn't appear to impact GRLevel3.
For remote access (outside my firewall) I use Himachi. Himachi (basic) is a free VPN software, by logmein, that connects one or more computers in a himachi network. I created a private network on himachi, which I join only my pcs. I then have full access to my remote computers like they are on the same local network. File Shares, Remote Desktop, and even my Itunes collection. Himachi appears as its own network adapter on your machine, and gets an IP address via DHCP.
I've been using it for a while now, and have not had a problem with security. I suggest using complex passwords, and using different passwords for machines/vpn/ and internet sites. I also recommend changing your passwords regularly. Its a pain but so is uninstalling trojans and rootkits left behind by a hacker.