If you just want to access the filesystem, set up an FTP server with FileZilla:
http://filezilla-project.org/download.php?type=server
You'll have to open up the correct port on your modem or router at home. This varies with modem/router models so I can't help you. The FTP protocol traditionally uses port 23, however some ISPs block common ports like that. Just pick any random number between 0 and 65535, the larger the better. Once you opened the port, run the server program, and configure it. It has a Settings menu or something, I don't have it here right now. Add a new user with a password and set up the paths. You can share your whole C: drive or just your user directory, or whatever floats your boat.
Then go to this website to check if the port is open correctly:
http://www.canyouseeme.org/
If it passes that test, then the hard part is over. Look for your IP address on that website. On your school computer, open up a web browser (or even Windows Explorer) or anything that can access an FTP directory and type this in the address bar:
ftp://69.279.176.62:4444
Assuming your IP address is 69.279.176.62 and you chose port 4444.
If you want to have complete remote control (like you were there) then you'll need something more crafty. If your laptop has Windows 7 Ultimate and the computers at your school have Windows 7 (any edition) then you can use Remote Desktop. However, the host has to be Ultimate. I find this an annoying requirement, but oh well. Just look it up. Otherwise, look into Symantec pcAnywhere (this was hype years ago, it may still be worth a try though) or GoToMyPC. I never used the latter, though.
Good luck!