Updated 19/2/21 (Thanks to Fshguy for the heads up this needed to be brought up to date with latest UI changes).
As of writing, you need to have played enough to have moved off of the lowest rank for at least one of the trading, exploring or combat ranks. For example, move from Harmless to Mostly Harmless in the combat rank.
See here for a rank overview:
http://elite-dangerous.wikia.com/wiki/Pilots_FederationOnce you have moved up at least one rank in any discipline you will be able to go to the Status console screen (hold
and
) and then use the dpad to select Galactic Powers (final option in the grid towards the bottom right of window).
This will bring up a screen that lists the Galactic powers. Use the dpad to navigate the various galactic powers available and hit
to open the power summary screen. Once selected, you can see an overview of the faction, rewards various bits of information. There is an option here to pledge allegiance at the very bottom of the screen in yellow with a little badge icon. Select that using
and the achievement will pop.
Once aligned with a power you can view more information about missions and ranking - although to get this achievement all you need to do is pledge, nothing more.
*** Spoiler - click to reveal ***If you align yourself with a power and then defect or leave you will likely be hunted for a while by your former power. There is a time limit to this, but expect more interdictions from faction AI. You also cannot rejoin that power after leaving for a period of time.