You can raise Fed Navy faction rep by checking your right UI panel (shift+D if you're on PC standard bindings), any faction listed there that is supporting Fed Navy will also increase your Fed rep.
So check your Bulletins in station, each mission will list who the faction is that gives the mission. Of course there are also Fed Navy missions but they're not as common. Also make sure you're doing missions in, or that go to, a Fed Navy base/system on the galactic map - it should be default, but you can filter to show which factions own which systems/bases. Supporting the opposing factions in a system is an easy mistake to make. Local system map helps.
And it's a bad thing to target ships that belong to a faction supporting Fed Navy - check your contacts when you scan, even if they have a bounty on them it will hurt your standing with their home faction. Dropping missions (including abandoning them because you're being paid to do so) also drops your standing.
And for the rank progression missions with the factions, check what goods they want you to get before you accept the mission - some of them are hard to get, or even illegal for Fed Navy. Those sly devils. Picking up the quest and realising afterwards will hurt you if you have to abandon it. With the way things go, the server population can change the loyalty of a system from Independent to supporting a faction - and if the majority of the systems in your part of the galaxy all belong to one faction then it gets harder to get illegal goods for these missions.