Motivation is going to depend a lot on whether you want them to be the good guys, the bad guys, or both. In story reality, there's (usually) very different motivations for a good guy org and a bad guy org, even if they do more or less the same thing. If they're the good guys, they have to do some unpleasant things sometimes, but their motivations are probably pretty pure, protect the world from the monsters, keep order, etc. etc. If they're the bad guys? They want to use the supernatural to dominate the natural. If you want a more nuanced approach with both good and bad? The "boots on the ground", so to speak, are good folks that want to protect the unaware and keep order, but the bosses are often unrepentant assholes who don't like agents that ask too many questions and sometimes the field agents wonder what the real aims of the agency are...
As far as department, determine what you want their motivation to be and slot them in where appropriate. If their main goal is observation and analysis, I'd say stick them in the CIA or NSA. If they're going the "protect the unawares" route, I'd say either DHS or a branch of the USAF would work best. Or you could make them part of the FBI in a way similar to what they did with Mulder in X-Files. I think my personal favorite way of doing it, though, is the way they did it in a Cthulhu supplement called Delta Green. It's a secret organization that has no official branch itself. Its agents are officially agents of other branches (ATF, DHS, CIA, FBI, Secret Service, sometimes even IRS) and spend most of their lives doing the normal stuff agents from those departments do. However, when something supernatural comes up in their area, or if they're specific skillset is needed, they get activated by DG by receiving a (usually barebones and vague) briefing. Doing it this way means that they're so secret that most of the government doesn't even know about them.
For operations, it'll depend a bit on where you put them in the government. If they're a branch of an existing agency (CIA, NSA, FBI, whatever) they'll likely operate similarly to that agency, with adjustments where appropriate for dealing with and covering up the supernatural. If you make them their own thing, you'll want to look at how other agencies operate and determine which policy structures fit your idea of the MIB the best. If you Delta Green it, operations are basically "handle this problem, don't fuck up, and lie like the devil on your after action reports".
