Here is a quick rundown on how to unlock the new weapons and gadgets in Battlefield 4 Dragon's Teeth. Please note that while you can now work towards these unlocks, you won't be able to unlock any of them until you play the Dragon's Teeth content as that is one of the requirements.
UNICA 6
These three are common across all unlocks:
- Reach rank 10
- Play a Dragon's Teeth map
- Dragon's Teeth

Award requirements
- Activate the floodgates on the Sunken Dragon map 0/1
- Get 5 kills while swimming 0/5

BULLDOG
- While on foot as Assault, kill 10 Support enemies who are also on foot 0/10
- While on foot as Assault, kill 10 Engineer enemies who are also on foot 0/10
- While on foot as Assault, kill 10 Recon enemies who are also on foot 0/10

MPXCS5
- 2 Sniper Rifle Ribbons 0/2
- 10 Radio Beacon spawns 0/10
- 20 Spot Assists 0/20

BALLISTIC SHIELD
- 5 Savior Kills 0/5
- 20 Suppresion Assists 0/20
- Destroy 3 explosives 0/3

ROAD WARRIOR DOG TAG
- 10 Capture Point Attacker kills 0/10
- 10 Capture Point Defender kills 0/10
- 25 kills with shotguns 0/25

DEAGLE 44
- Get 20 headshots with the Unica 6 0/20

REMOTE CONTROL DOG TAG
- Destroy a vehicle with the EOD Bot 0/1
- Get 5 kills with the RAWR 0/5
- Get 10 kills with the Mortar 0/10
- Spot 20 targets with the MAV 0/20

KILLJOY DOG TAG
- Get 10 kills while under 30 health 0/10
- Break 30 enemy links in Chainlink 0/30
- Win a round of Chainlink with 50 tickets left 0/1

CITY SLICKER DOG TAG
- Play 10 hours of Dragon's Teeth 0/10
- Get 100 kills on Dragon's Teeth maps 0/100

There is also the new Phantom Initiate assignment. The password required to unlock this assignment has not yet been revealed. However, the requirements for this assignment have been discovered. You may recall the method for inputting the password for the Phantom Trainee assignment.
PHANTOM INITIATE
- Get 200 LMG kills
- Get 50 kills with explosives
- Get 25 roadkills

This will unlock the Phantom weapon and vehicle camo.

If and when the password is discovered for the Phantom Initiate program, we'll be sure to let you know.