Warframe

Disclaimer: This achievement isn't possible until after you've completed the Second Dream quest.

To build a Railjack, you'll first need access to a Dry Dock. Check out the achievement below to learn how:

You'll then need to go to the Dry Dock and you or your clan will need to research the Railjack Cephalon blueprint at one of the central kiosks there. It takes 1 hour to research.
Make sure to duplicate the blueprint for 35k after it finishes researching.

Ghost Clan x1, Shadow x3, Storm x10, Mountain x30, Moon x100
Credits: 10k
Mutagen Mass: 5
Detonite Injector: 5
Fieldron: 5
Orokin Cell: 4

Then you'll need to go to your Foundry on your Orbiter to build the blueprint. Here is that individual cost:
Credits: 25k
Mutagen Mass: 5
Detonite Injector: 5
Fieldron: 5
Orokin Cell: 2

After you receive Cephalon Cy (the Railjack Cephalon), it will immediately give you the quest Rising Tide.
There are 6 derelict parts of old railjacks you'll need to track down and repair to complete your new ship.
Check the wiki here if you need help with the quest:
https://warframe.fandom.com/wiki/Rising_Tide

Edit: These parts used to be very resource expensive but are much easier to build now.
Each of the 6 parts need to be built in the Dry Dock and each are very resource intensive. Each player has their own ship so you're the only one that can contribute resources to your railjack; your clan can't help. Also, each takes 12 hours to complete and cannot be rushed using Platinum.

Total cost to repair all 6 parts:
Credits: 6,000,000
*Cubic Diodes: 100
*Carbides: 120
**Copernics: 120
**Pustrels: 100
Argon Crystal: 15
Orokin Cell: 30
Neural Sensors: 30
Neurodes: 35
Morphics: 50
Control Module: 200
Rubedo: 15,000
Circuits: 5,000
Polymer Bundle: 4,500
Plastids: 3,000
Cryotic: 1,000
Alloy Plate: 27,000
Ferrite: 20,000
Salvage: 10,000
Detonite Ampule: 500
Fieldron Sample: 500

***All 4 of the new resources drop in large quantities from public Railjack missions. If you'd prefer not to do Railjack missions until after you have your own built then check out the 2 paragraphs below to find out how to farm them in other missions.

*New resources - Cubic Diodes & Carbides only drop from Eximus enemies; CD's only on Eurpoa Ice Planet tilesets and Carbides only on Ceres Shipyard tilesets. I prefer to farm them on Dark Sector nodes for the innate increased resource boosters. This means Cholistan & Larzac on Europa for CD's or Seimeni on Ceres for Carbides. Eximus enemies appear much more often on later rounds. I had over 150 of each after just a single 25 round Defense using my Smeeta Kavat.

**New resources - Copernics & Pustrels are mined in red veins on Orb Vallis & Plains of Eidolon respectively. The spots that tend to be most heavily laden with ore veins that I've found are Deck 12 on Vallis & the Twin Horns cave on Plains.

After repairing all 6 parts of the ship you'll FINALLY be able to enter your new Railjack! Congrats!

With the start of the Update 30: Call of the Tempestarii, it is now possible to purchase a fully-built Railjack in the in-game Market for 400 Platinum. Obtaining the Railjack this way will count the Rising Tide quest as completed.
Usually a Railjack is constructed during the Rising Tide quest, which requires completion of The Second Dream.

In my case (purchased the RAILJACK CAPTAIN’S BUNDLE), I had to replay the Rising Tide quest because the Achievement didn't unlock. If you should encounter the same problem, just replay the quest as described. The quest for itself is pretty straight forward and can easily be completed within 1,5h because - due to the in-game purchase - building and repair times for the diverse components of the Railjack are no longer required.