As Itaki says, you have two things to level-up with your Major Faction; Reputation and Rank. They both increase as you complete the right missions.
'Reputation' seems a little incidental, I've not noticed what effect it has on the game.
SCREENSHOT
Increase your 'Rank' % by completing missions for Major Faction affiliated factions. When your 'Rank' reaches 100%, you will be offered missions with 'Imperial/Federal Navy/Naval' in the title, and something like 'Promotion to [Outsider]' in the reward description. On completion your Rank will be back at 1 or 2% but you'll be a rung further up the ladder -
http://elite-dangerous.wikia.com/wiki/Empire/Ranks
Promotion missions seem to be worth slightly more Rank % increase than other missions so they're worth stacking if you get the opportunity, but don't expect to get 5 promotions from stacking 5 promotion missions.
And as SirOleg says, if you're in the right system with the right factions and have Rank 100% already, keep going with standard missions and the % will be added when the Promotion missions finally turn up, so you don't need to go hunting for them.
The more minor factions at your station that are affiliated to your Major Faction, the better.