Simple question, I want a door to be unlocked when I finish talking to a certain character about something.

However I am unsure of how to do this. Is there some way to just type this command into the text box in the dialogue menu? Or do I have to edit the Script Fragment with Notepad++ like I do with other scripts?

You should be able to use the Papyrus fragment labelled End in the Topic Info window to unlock any reference by adding an ObjectReference property to the fragment and typing something like the following into the End fragment:

DoorRef.Lock(False)

Or you can have the same fragment advance the quest to another stage, which in turn has a script fragment do basically the same as above. There's probably more ways to skin this cat.

Personally, I would use the quest stage method. That way if a user had something misfire with the dialog fragment, you can still set the stage with the console so they're not stuck.