The recruitment process
Obviously, you need an organization in mind. Your first step is to join that group to whatever extent is possible in the game. If there is a reputation, raise it; if there is an actual group to join, join it. Yes, this may mean that you have to do something you're not totally thrilled about doing; I can completely understand looking at a reputation climb in World of Warcraft or Star Wars: The Old Republic and feeling the wind go out of your sails. But if you're going to sign up, you're going to need to sell it at least a little.
Sometimes this is easily done. If you start playing an Illuminati character in The Secret World, then congratulations: You've joined the Illuminati! Other times it's difficult bordering on impossible; good luck passing yourself off as Phoenician in the same game. The point is that you identify the group and do your best to look and act like the rank and file.
And yes, looking the part is important. In some games it's trivially easy to replicate the outfit of your chosen group. Star Trek Online uniforms can easily be shifted to match that of any branch of Starfleet, for instance. Games like Final Fantasy XIV that currently lack a vanity slot might make this a little bit harder outside of specific roleplaying outfits. Don't worry too much about an exact match, though; just make sure you look close enough that your outfit is recognizable.
Benefits of membership
The great thing about being in any sort of organization is that suddenly you have a readily accessible reason for your character to do literally anything because all you need is an order from the top.
Can't figure out why your character would care about a given in-game event? She was ordered to take part. Not sure why she sticks around in a guild from an IC standpoint? Ordered. You want to start a new plot for her that requires her being in a specific spot in spite of her lacking a reason to go there in the first place? You get the idea. Arbitrary orders cover a multitude of plot points.
More than that, though, being a part of an organization can also push characters in new directions and offer them new sorts of responsibilities. No matter how irresponsible you are when you start, when you're working with a team, you tend to learn some responsibility or quickly get fired. That gives you opportunities to meet new people and exposes you to new situations, giving you some crucial chances for character growth that you might otherwise miss.
From a more meta standpoint, organizational membership also creates a sense of verisimilitude (down a shot) in that these characters aren't simply moving on a stage. These are people living in a world, and these groups make a difference. You probably see it in real life; your friend Phil is a volunteer firefighter, Nancy was in the Army Reserve in college, no one can get Dana to shut up about working with the Salvation Army, and so forth. These groups aren't names; they're collections of people.
Behavior in the ranks
Every organization in the world has a hierarchy. Even if the organization consists of just four people living in the same house, someone comes out on top and someone's at the bottom. This means that going up in that hierarchy can be a major point of character development, and your character can slowly rise from being irrelevant to a position of theoretical power.
There are two problems here (well, two and a half, really, but we'll get to the half in a moment). The first is that normally increased rank comes with increased responsibility; the second is that the power you wield is largely theoretical. You can be a Jedi Master in Star Wars: The Old Republic, but you can't reliably have other Jedi jump to your tune.
Military organizations in games usually get around this through the chain of command. Of course you can't just randomly order soldiers to do something; they're assigned to someone else. In-game, you're generally going to have to accept that your character's power makes a better threat than an immediate trump card. You can't have two dozen soldiers ready to show up on your order, but you can ensure they will be there pretty quick.
Increased responsibility is a different matter. You might hate your boss at your job, but odds are near absolute that he or she puts in a lot of extra hours that you don't and has a lot of responsibilities you don't see. (Yes, reading this article is probably something your boss doesn't have time to do. [Except when you're Eliot's boss. -Bree]) There's no real way to reconcile the responsibilities you should have with your increased privilege, but the best option it simply alluding to the idea that your character's time is a bit more constrained.
Or perhaps the reason your character keeps getting assigned to horrible things she doesn't want to do is because she never fills out her forms and doesn't handle all the responsibilities she's supposed to fulfill. It's a distinct possibility.
The "half" problem I mentioned above is that... well, sometimes promotions aren't exactly an in-character thing. Either there's no way of modeling earning new ranks or new ranks are earned with trivial ease. In both cases, a certain amount of careful judgment is necessary. You are not Supreme Allied Commander after two weeks, and you're not going to still be working in the equivalent of the mailroom after a year of exceptional service. Have a care.
Feedback is welcome in the comments below or via mail to firstname.lastname@example.org. Next up, it's time for fear; the week after that, I'm going to just write your next character story for you.
Every Friday, Eliot Lefebvre fills a column up with excellent advice on investing money, writing award-winning novels, and being elected to public office. Then he removes all of that, and you're left with Storyboard, which focuses on roleplaying in MMOs. It won't help you get elected, but it will help you pretend you did. If you need a refresher, check out the Storyboard Library.