This is more of a helpful tip I tried (which worked) until a proper guide comes along for those that already had collected/listen to 5 answering machines messages before the April patch and never got the achievement or that haven't got it yet.
If you already have them collected then look in your mission log for missions that have a cassette tape icon (not all cassette icons are answering machine messages). When you click on the cassette look for this picture:
1. Load your game and go to your log with all your missions listed.
2. Scroll through your missions and if they have a cassette tape icon, click on the mission and click the cassette tape icon.
3. After clicking on the cassette tape icon (and it has the picture above) then play/listen to the recording.
4. Do this to 5 different messages and the achievement should pop (if not sooner).
For those that are looking for or already have them here is a list of quests with answering machine messages:
The Road to Salthamn
Calling for Help
Seeds of Doubt
Beyond the Barricade
The Girl Who Cried Wolf
Stealing from the Poor
The Gas Factory
Science of Destruction
There maybe more but this is what I have in my mission log with the answering machine picture and is more than you need for the achievement. Good luck to everyone and hope this helps out anyone trying to go for this achievement.