How to get to the trail from New Orleans without a car?

That's a fairly big ask, but not impossible. The Black Creek Trail's eastern terminus starts from Fairley Bridge Landing, which just so happens to be near the small town of Brooklyn and a well-regarded local outfitter called Black Creek Canoe Rental. They actually put on a shuttle for paddlers but which is also open to hikers who want to hike the trail. More info here: http://www.blackcreekcanoe.com. Driving to Brooklyn from New Orleans takes a little over two hours, but you can also ride Amtrak's Crescent City service between N'awlins and Hattiesburg, MS. From Hattiesburg, it'll probably be about a half-hour to Brooklyn by cab, or you could just have them drop you at the trailhead itself for a slightly higher fare. Not sure how this works for you, so map it out and see if it may be cheaper or more convenient to rent a car. You might also just call Black Creek Canoe Rental to see if they have any tips -- hey, they might even know some local hikers who are heading up the same time you're planning your trip! Have fun. By all accounts, that's a pretty special trail for backpacking!