For this achievement you need to recruit 5 DIFFERENT companions from across the Boston Wasteland. Simply recruit the character then dismiss them and move on to the next one. I apologise in advance that I do not have a video to show you each companion's location, but I'm sure you can find them by yourselves! :)
There are 13 companions that you can team up with:
Cait - Location: Combat Zone
Codsworth - Location: Sanctuary
Curie - Location: Vault 81
Paladin Danse - Location: Cambridge Police Station
Dogmeat - Location: Red Rocket Truck Stop
Deacon - Location: Old North Church
Hancock - Location: Goodneighbour
Maccready - Location: The Third Rail (Goodneighbour)
Piper - Location: Diamond City
Preston Garvey - Location: Museum of Freedom, then moves to Sanctuary
Strong - Location: Trinity Tower
Nick Valentine - Location: Vault 114, then moves to Diamond City after quest
X6-88 - Location: The Institute
Many have quests attached to them that you must complete before they are available to become a companion, which can be a pain if you want to recruit them quickly. Therefore, by far the easiest ones to recruit are:
Codsworth, Dogmeat, Preston Garvey, then Piper and Nick Valentine (once you get to Diamond City). However, it is completely up to you.
Good luck!