To rank up for any faction you need to take Navy missions. To get these to appear you need to work for the minor factions. Only ever do missions for Empire minor factions (ones with the symbol at the end). Then wait for good luck to hope for a rank mission to appear (as this bit is luck).
Work your way through lots of different minor Empire factions, working for each one. I think Tun is a good area, as there are about 2 or 3 systems locally you can work on. Basically get to Allied with a minor faction and move on to the next.