"Se faire" can have two meanings.
The first one is used in a sentence where someone does something to the subject.
Paul se fait battre par John.
Paul is being beaten by John.
In this case, Paul is the subject of the sentence, but he is the one who is recieving the beating. In can be compared to an English present continuous passive sentence (how twisted!), the passive of "John is beating Paul".
But there is a particular meaning, when the subject of the sentence is the one who does the action.
Je me fait petit pour éviter d'être vu.
I make myself small to avoid being seen.
In this case, "se faire" means "to make oneself" or "to turn oneself into". Literally or fuguratively. You can say "La chenille se fait papillon" (The caterpillar turns itself into a butterfly).
Back to your example:
Je fais l'avocat du diable.
I do/play the devil's advocate.
And
Je me fais l'avocat du diable.
I make myself the devil's advocate.
Both are correct and used.
As the writing is similar, we usually don't see the difference in French, especially when speaking. But "I make myself the devil's advocate" implies that you are trying to look like one, even if you are not, pretty much like "I make myself big" of "I make myself strong".