For those who don't want to read this guide. Here is a video that summarizes what you have to do to unlock the Phantom Bow and Phantom Operative Assignment.

Premium Membership for Battlefield 4

Battlelog linked with your Xbox Live/PlayStation/Origin account
Three other friends or players to do this with

First things first, you will need to complete three Phantom assignments in order to unlock the Bow. The three assignments that will need to be completed are:

Each Phantom assignment will need to be unlocked via Battlelog before you can start progress on each assignment.

Phantom Prospect is the first Phantom assignment that was added with the China Rising Map Pack.
In order to unlock this assignment, you will need to log into Battlelog and go onto the leaderboards map. In the bottom left hand corner of the map there is a skull that you will need to click onto. A console will open on screen and will

start showing a bunch of text. Once it has finished loading the text you will need to enter the following code into the console: bumpinthenight
After you have entered the code you will be granted access to the Phantom Prospect Assignment.

In order to complete Phantom Prospect you will need to complete the requirements which are:

These are pretty strait forward requirements that don't require a lot of effort to complete other than time.

Score 200 kills with Main Battle Tanks
To narrow down what tanks count towards the assignment are the following:

You can also use the tanks secondary mounted light machine gun for kills and it will count towards the progress.

Once the requirements has been met you will unlock the Phantom Program dog tag

Phantom Trainee is the second phantom assignment that was added with the Naval Strike Map Pack.
Just like the Phantom Prospect Assignment you will need to unlock it through Battlelog before you can start completing the requirements. To unlock this assignment will be the same as the Phantom Prospect Assignment except you will need

to enter a different code into the console. The code for this assignment is the following:
epic dream worlds
After you have entered the code, you will be granted access to the Phantom Trainee Assignment

In order to complete this assignment you will need to meet the following requirements:

Again these requirements seem pretty strait forward except they need to be done in a certain order.
The pistol kills and jet kills requirements need to be completed before you can start and complete the shotgun ribbons requirement.

Score 20 kills in a jet
The easiest way to complete this requirement is by playing Air Superiority.

Get 2x Shotgun Ribbons in a round
In order to get a shotgun ribbon you need to get 6 kills with any shotgun so a total of 12 shotgun kills is needed in a single round.

Once the requirements have been met and completed, you will unlock the Phantom Weapon Camo/Paint.

Phantom Initiate

Phantom Initiate is the third Phantom Assignment that was added with the Dragons Teeth Map Pack.

Again to unlock this assignment, you will need to unlock it through Battlelog before you can start

completing the requirements. You will need to enter a code on in the console on Battlelog the same way as the previous Phantom assignments. The Code for this assignment is the following:
24344241893
After you have entered the code you will be granted access to the Phantom Initiate assignment.

The following requirements will need to be met in order to unlock this assignment:

Again these requirements are fairly easy and strait forward.
As far as I am aware, these requirements don't need to be completed in a certain order.

Get 50 kills with explosives
Some explosives don't count towards this assignment so these are the explosives that do count:

Get 25 road kills
There are two ways this this requirement can be done, either through using road vehicles to kill players or to use the SUAV and fly it in to players.
I found the SUAV to be the easiest and the quickest as it can be reused from your gadgets every minute or so.

To unlock the SUAV you must complete the Safe Raiding assignment.

After you have completed the requirements to this assignment, you will unlock the Phantom Body Camo/Paint.

Final Stand Dog Tags

The Final Stand Dog Tags were added with the Final Stand Map Pack. They are a requirement in order to unlock the Phantom Bow.
You will only need to find one dog tag from either of the maps to unlock the Bow.

If you are doing this with three other friends/players you will need to make sure that you each find a different dog tag to equip at the end.
Example: One player has the Hangar 21 dog tag, another player has the Hammerhead dog tag, another player has the Operation Whiteout dog tag and the last player has the Giants of Karelia dog tag.

These are the four following maps that were added with the Final Stand Map Pack:

There are a total of four Dog Tags that have been hidden across the four Final Stand Maps.
Only one of the dog tags can be found on each map but can be found multiple times within the same game.
It must be a ranked game and must have the amount of players required for the game to start.
So far it has been known that a dog tag will spawn 5-10 minutes after the game has started and will respawn 7-10 minutes after a dog tag has been found and picked up.

Disclaimer: I take no credit for the following images or videos.
Credit to _HJP_ for the images.
Credit to Blindman for the videos.

Hanger 21

The following image shows the locations of the Hangar 21 Dog Tag.

The following video will show the locations more in depth.

Hammerhead

The following image shows the locations of the Hammerhead Dog Tag.

The following video will show the locations more in depth.

Giants of Karelia

The following image shows the locations of the Giants of Karelia Dog Tag.

The following video will show the locations more in depth.

Operation Whiteout

The following image shows the locations of the Operation Whiteout Dog Tag.

The following video will show the locations more in depth.

Conclusion to the Above
Once you have completed all three Phantom Assignments and have found your self a dog tag from either of the Final Stand maps, you will need to do the following to progress to the next step:

Hanger 21 Elevator

Once you and your other friends/players have done the above, you will now be able to access the Phantom Bow room by using the elevator on the Map Hanger 21.

The Elevator can be found on Hanger 21 at the far top of the map at the A Flag on Conquest Large. It will be on the lowest level underneath the A Flag to the right which is also at the end of the tunnel that is to the right

of the A and B flag.
You can use the image below for reference.

In order to use the elevator to go down to the Phantom Room, all players will be checked in the elevator to ensure that they have met the requirements of the conclusion Above. If all players have met the requirements, the button for the elevator will turn green and can be interacted with to use the elevator. If a player has not met the requirements then the button will not turn green and wont be able to be interacted with.
Any players that manage to spawn inside the elevator as it is going down and don't meet the requirements will be killed automatically by the elevator.

Phantom Bow Room

Once you have taken the elevator down to the room, you will see a force field and a keypad to the right of the force field. The keypad is used to unlock the force field.
Only one person can use the keypad to unlock the force field for all players. The code for the keypad is the following:
1290-429-397648-970
If the code is input correctly, the force field disappear and you will be able to walk around in the room and pick up the Phantom Bow.
Once you have picked up the Phantom Bow, it will then be available to use on all classes as a Carbine.

Phantom Operative

Phantom Operative is the final Phantom assignment that is unlocked from the Phantom Bow Room.
In order to unlock this assignment you will need to interact with some documents on the right hand side of the Phantom Room. They will be in top of some filing cabinets that you will need to jump onto to interact with them.
Once you have interacted with the papers you will be able to start progress in this final assignment.
Currently Battlelog is glitched with this assignment and doesn't show the requirements as you can see in the image below.

You can still complete the assignment as the assignment is view able on Battlelog on Xbox/Playstation.
The requirements for this assignment are the following:

Once the requirements have been complete you will unlock a Dog tag which can be seen below.

So after seeing a few threads of people asking to squad up to get into the Bow Room, I thought I would add how I got the Bow without having friends/Squad to do this with.

Once you have completed all assignments and found a hidden dog tag and have everything required to get into the bow room, find yourself a 64 man server on Hanger 21 that is specifically for finding dog tags or going into the bow room.
When on the server, go to the elevator and wait inside it until people come over to also get into the bow room.
(There can be more than 4 people in the elevator as long as they have every requirement)
From here you will just have to keep waiting and trying until people get in the elevator and the button turns green.
Eventually you will have enough people in the elevator to go down and get into the Bow Room.

Note
This can take some time so you will have to be patient.