The achievement is for finishing all of the following campaigns on expert difficulty:

• Dead Center
  
• Dark Carnival
  
• Swamp Fever
  
• Hard Rain
  
• The Parish

A campaign has always registered for me when I completed it from start to finish, but if I joined a match in progress then it only has half a chance of contributing to the achievement. It seems that you need to survive the rescue for it to register.

You can see your achievement progress from the Steam library > Left 4 Dead 2 > Achievements (you may have to scroll down to find the "View My Achievements" button). This will show the progress in number.

But the in-game achievements list found in the menu has a dropdown box that can be opened for the achievement which shows the ones that have registered as completed. Progress might update slowly, but the achievement itself should show up instantly when you finish your last campaign of those 5.

With a combination of skill, good teammates, and some basic tactics for dealing with difficult bits, it shouldn't be too difficult to get this achievement. If you're struggling to complete a campaign on expert then consider warming up with a playthrough of it on a lower difficulty, researching how to tackle any parts you're stuck on (perhaps by skimming through a YouTube video of it being done on expert), and try to find teammates that are as motivated to finish it as you are.

I've read that it's possible to get this achievement using mutations to help make it easier, so if you're stuck then that's an option to research.