First you need build up some reputation with a minor faction at the outpost you work out of. Do a few simple trade-runs. Won't take much. When you check the Bulletin Board, what you are looking for is the missions that tell you to go to system X and pick up items Y and that the activity may be considered illegal. Then jump to system X, and visit every USS that pops up. Usually you'll find what you are looking for within 3 tries. Be it black boxes, ancient art, prototype tech, rebel transmissions or whatever else it might be. Pick up as many as you can. Then head back to where you got the mission, hand over the illicit goods by visiting the Bulletin Board again, and then give cargo. Any surplus cargo you sell on the black market, which you find under contacts.
Do note: Once you've picked up such cargo, it will be marked as stolen. So do -not- go visiting any other USS or base or anywhere else where System Authority Ships may scan you. Go straight back to your home outpost and get rid of the cargo at once. No detours, no stopping by old friends doing some side-trade and catch up on gossip, no break for a cafe-latte with cream on top. The very last thing you want is to get caught red-handed with stolen cargo in your hold.