You'll need to edit your Logbook file in logbook.lua, which is usually found in C:\Users\*UserNameHere*\Saved Games\DCS\MissionEditor.
You might have to download Notepad++ to open it (if I recall, it's an unformatted mess without it), but make a copy (or 3) before you try anything. I don't remember the specifics, but you should be able to see successful missions from the same campaign to see what you need to add and change. Add the relevant information, save it, load up the campaign and see if you've advanced. If not, copy one of your backups over and try it again.
It wasn't too complicated to figure it out. If I did it, you'll be able to. If you really don't want to mess with it, you should be able to play the campaign missions as individual missions.
I love that campaign, by the way.