Even though I'm new, I just spent a lot of time and money grinding Federation Rank up to Rear Admiral. (so I could get a Corvette myself), so I may be able to help a bit.
First: the skimmer mission malfunction is apparently a separate issue, experienced by many people. (Reference
https://forums.frontier.co.uk/showthread.php/254302-No-Skimmers-at-destroy-skimmers-mission-location , et al.)
Second: ensure that the clans you do missions for are in fact Federation clans, and have the seal showing, like this:
Third: After you complete missions for Federation clans, in the status screen on the right, under Reputation you should see your Rank bar progress:
Fourth: if you are willing to travel out to the CEOS system, you can usually grab several data delivery missions at once to the nearby Sothis system (more if you log back and forth between Open and Solo). It doesn't even require any cargo space. I think the game will let you stack up about 20 missions or so, and then the trip to Sothis takes about 2 minutes, and you can cash in all those missions at once. (If you only take missions to the Sothis Mining station, you won't have to make two stops.) Then you can pick up many more data delivery missions back to CEOS. Rinse. Repeat.
Fifth: If you really feel like being lazy, and you have some serious bank, and you're willing to travel to 17 Draconis, (once you've established a minimum rep) you can just sit there and do donation missions all day. Rear Admiral took me about 3 days and 250 million credits. But again, only two of the four clans there are Federation affiliated; don't bother with the other two.
Sixth: Once your rank bar says 100%, now you want to look for missions that say the exact words "
Federal Navy" somewhere in the title. (If at 17 Draconis, you'll have to leave that station to get that type of mission offered; 17 Draconis almost never offers them. I recommend Tun with 5 Federation clans, to increase the likelihood one of them offers you a Federal Navy mission. If you don't see one on Open, check Solo.) After you complete
that mission, then you will be awarded a higher rank.
You don't need to do the rank-up Federal Navy mission right exactly when your rank bar reaches 100%; any additional Federation missions will carry over onto the rank bar of your new rank. But
do make sure it's 100%+ and not 99.6% rounded up to 100 when you go looking for a Federal Navy mission.
Finally, you may get messages saying that you have achieved a higher rank than your Rank bar says. This happened to me and many other people and is (supposedly) a known issue (although it's not listed on the known issues list.) Just ignore those messages.