How to curry favour?
Firstly, don't do anything that'll make the factions mad(ras) at you. Once you cross them, you get a fine or even a bounty on your head. Korma's a pickle mistress.
But seriously...
I haven't done many missions myself, but there are non-mission ways to help this.
Fighting in general seems to help. Fly to a RES in a system controlled by the ruling faction, especially a RES with authority ships fighting there, kill Wanted folk, hand in hundreds of thousands worth of bounty vouchers to the appropriate faction. For me, I very quickly rose to Friendly to the major factions on an interstellar basis.
Also, handing in exploration data will help, but it seems a fair amount will need to be handed in.
Finally, trade regularly with that station and the faction will slowly like you.