Edit: People are reporting a Black Mamba error when purchasing a carrier, and when they log in, the carrier is there but the achievement is not. The best I can think of is to first submit a ticket to Frontier, but also perhaps to decommission the carrier, then buy it again after the week cooldown.
Just spend 5 billion at one of the starports that sell them.
Since money can't (easily) change hands, you'll likely have a firm grasp of the game by the time you get this, so I'm just letting you know what you need in case you don't know. The good news is that if you don't intend to keep it, you can sell it and get almost all of the credits back after the 7-day cooldown.
Edit: Thanks to xRosierrox for pointing out that I should post the list of stations that sell them.
Colonia – Jaques Station
Ehlanda – Clark Port
Agartha – Enoch Port
Panoi – Garan Hub
CD-47 990 – Jackson Ring
Lan Tzak – Jacobi Platform
Namnetes – Jolliet Enterprise
Kruger 60 – Kepler Gateway
Skeggiko O – Kuo Terminal
Balante – Laplace Ring
Alcor – Macdonald Settlement
Kakmbutan – Macgregor Orbital
Diso - Shifnalport
You can purchase it at the Brewer Corporation Carriers contact.
You can also see a list on Inara:
https://www.inara.cz/galaxy-nearest/28/367/0/You can outfit your carrier at a number of other systems. These aren't apparent in game, but they're also listed on Inara:
https://www.inara.cz/galaxy-nearest/32/367/0/...and near Colonia:
https://www.inara.cz/galaxy-nearest/32/77690/0/