I see in the rules what to do when Duty increases to 100, but how do you actually increase Duty?

It's a roleplaying thing. The GM should craft situations that incorporate the PCs different Duties, and then the players should roleplay out their duties, and at the end of a session/adventure/campaign, the GM raises their Duty as he deems appropriate. If you picked up Onslaught at Arda I, it periodically has boxes that mention good spots in the adventure where a PC's Duty could come into play. You can also double check Operation: Shadowpoint, as it might have similar boxes. Also too, at the end of major adventures/campaigns, providing the group is successful in their missions, Duty can rise as well. I recall something about not being stingy with how much Duty is awarded either, but not where (i.e., not 1 or 2 at a time, but 5 or 10). A developer question or something.

It is, like the other key mechanics, dependent on the entire group to actively work towards.

As Blackbird said, but to expand on it a little. It's earned like XP; you look at your PCs Duty types and see if anything they did would match with their type.

Typically, if the PCs did a mission for the Rebellion but nothing specifically tied in with their Duty, I'll give that PC 5 Duty. If something did tie in with their duty, I'll award them 10. Now, this is for a group with 5 Duty-contributing PCs, so that means they typically reach a Commendation Rank every 3 sessions. Considering my sessions are 8 hours long and played every 6-10 weeks, that's not too bad a progression, I feel. If you play 3-4 hours every week or every other week, they'd go up in rank every month at that rate. If you play more frequent, make the token award 3 and the specific Duty award 5-7 and that should work out.