How can I increase my FAF rating?

Contrary to common opinion, there is no complex algorithm looking into who killed the most ACUs, your mass killed, your resources produced, or anything like that. All that matters for your rating is:

1. Is the game ranked
2. Did you lose or did you win
3. What are the ratings of you and your opponent

This means that if you were playing a teamgame where you died early but your team won, you get the equivalent points of the player that carried the game. Likewise, you could have played an extremely intense hour long match against somebody or simply lost at minute 3. It's all the same to the system.

The most efficient way to gain rating is by beating players that are better than you, particularly when there are less total players in the game. Why is that?

When you introduce additional players into the equation, the system has a harder time gauging your ability as a player due to the inherent variability of additional people. It cannot be sure if you were particularly capable or if your team won the game. This means it can give less weight to the end result which in turn leads to you gaining less rating. On the otherhand, if you consistently beat 1800s in 1v1s, there is nothing that can cause the system to question whether you truly are better than an 1800.