If you're looking for specific, high quality doffs, most of them can be found on the exchange. But if you're just looking to do some roster building, here are some tips:
The personnel officer at the academies (Starfleet Academy in the building with Lt. Ferra and Cmdr. Valia, Klingon Academy next to the exchange, and Romulan Republic command [this one is a console in the room with the bridge officer trainer]) have free missions that can be done every couple of days that award doffs. Beware the General Recruitment one because it costs dilithium, as does the Request R&D assistance (though that one is usually worth the dilithium cost). The other 4 missions though award 2-3 doffs upon completion. These are usually white quality, sometimes green (though I did once get one of the purple original ultra rares from it).
Refugee missions are great for roster building as well. Buy up some common refugees from the exchange, and then grant them sanctuary/refuge via the available doff missions. If you are playing a fed character, buy the doffs Neva something or other and Jeffery Toby Lamar. They have all 3 necessary traits for the mission, which means it will often trade the common refugee for Blue/Purple doffs. The mission is available at ESD under the ops tab, as well as in orbit of Sol System. The mission repopulates about every 36 hours, so you can do this fairly consistently to get some good doffs. If you go out in to the Sirius sector block, you'll also find grant Asylum on Vulcan/Andoria/Rigel V/Tellar Prime, there are doffs that line up well with all of these as well, but I can't remember their names offhand. If you're KDF, it becomes even better, because the Omega Leonis block/Qo'nos/First City maps almost always have 1-4 missions available that are of the grant asylum/sanctuary/refuge type that will grant Klingon, Nausicaan, Gorn, and Orion doffs.
All of the exploration clusters have a 7 part mission chain that ends by awarding a unique blue doff. Then, once those have been completed, they have a repeatable mission that on success awards a refugee (fed) or prisoner (kdf) and on critical success award a purple version of the blue doff you get at the end of the chain. (B'Tran cluster at the bottom of Gamma Orionis block awards liberated borg technicians [caitains for fed, gorn for kdf])
Doff packs are a pretty good way to grind too. Reinforcements, Fleet Support, and Romulan Surivivor duty officer packs can be found on the exchange and award 7 doffs, with special powers. These occasionally drop a purple doff, and are guaranteed to drop at least one blue doff.
Special doff requisition packs, like the Elachi Survivor, Voth Separatist, Tholian Warfare Specialist, and Dominion Duty Officer, reward 1 (2 in the Elachi pack) purple doffs with unique powers. They're pricey, but these doffs usually have traits that are good for many of the doff missions out there (most of these doffs have resolve, which is a useful trait).
Also, if you talk about doffs in the fleet chat when I'm on, I'll often mail you a bunch in the in-game mail (I have thousands of doffs and don't need most of them, so I like to help out where I can B) )
If you have any questions, there are lots of resources out there, but hopefully this gives you a little bit of an intro.