I had some issues in trying to get the DLC to begin, so this solution will address that if others experience the same issues.
After installing the DLC (it appears as 'DLC3' in the Manage Game section for Watch Dogs 2), the Moscow Gambit may be listed in the DedSec App as 'DLC Operation' with the text "Operation to be unlocked". It states that you must complete the 'Hacker War' operation in order to unlock Moscow Gambit. The issue is that it will state this even if you have already completed Hacker War.
If you have this same issue, simply select the Hacker War operation from the DedSec App, select the 'Bunker Bust' mission and choose to replay it. Once you complete this fairly short mission, you should finally be contacted to begin the Moscow Gambit operation.
EDIT: MathNotEvenOnce says that it is actually the first mission of Hacker War that you must replay. If that is the case, then the solution still holds but you must instead replay the first mission. If you are still not contacted to start Moscow Gambit, then you should replay Bunker Buster as well. In either case, replaying the mission will allow you to start Moscow Gambit.
The Moscow Gambit operation itself consists of 5 fairly short and fun missions, which are:
1. Cheeky Bastard
2. Under Pressure
3. Corrective Measures
4. Cripple the Bratva
5. A Grand Finale
Once you complete the finale mission, enjoy the fireworks show and your 40G achievement.