The hardest part of this achievement is going to be unlocking the weapons.
For the MAC-10 you will need to complete the Professional Assignment 2.
To unlock the assignment you will need to get to rank 7 as a professional and complete Professional Assignment 1 which involves getting 10 kills with snipers and 5 kills with the laser trip mines.
Once you unlock Professional Assignment 2 you will need to complete it by getting 25 kills with snipers and 5 camera coins, which are 10 enemies killed while spotted with your camera.
The easiest way to get the camera coins is to play Conquest Large on Dust Bowl. If your team has captured the "B" point then place cameras on the windows pointing outward towards A, C, D, and E. If you don't have B yet point them inward towards B. Most of the fighting seems to happen around B so if you stick with it you will eventually get the coins. It took me 3 games to get 5 camera coins.
Now you just need to get 25 kills with your new MAC-10 which is considered a secondary weapon for the Professional class.
For the Double-Barrel Shotgun it will take a considerable amount of work. To get the gun all you need is the Enforcer Syndicate Assignment. Easy right? Not so fast. To unlock the assignment you will need to complete Enforcer Assignment 2 which is the enforcer class' equivalent to the professional one needed to unlock the MAC-10. You will also need to unlock 10 enforcer weapons or gadgets, and get Enforcer Gold Service Star 1. All that is just to unlock the assignment to start working towards that.
To actually complete the assignment and unlock the shotgun you will need to get 200 shotgun or Battle rifle kills, unlock the Enforcer's tier 4 reputation track in 10 different matches, earn 15 ammo resupply coins, and finally get 15 ballistic shield coins. OK! Finally after all that you will unlock the Double-Barrel Shotgun and now the 25 kills should be an absolute breeze after the hassle that unlocking it was.
According to DJ KOOLAIDE if you know somebody that has these guns then they can drop them and you can get kills with them and they will count towards the achievement. Might be a good idea to keep an eye out for enemies using these guns as well. It would probably be quicker to kill them and take the guns.