Route 61 follows the line pretty closely, but alas the tracks are rarely in sight from the highway. This is a particularly unattractive part of Louisiana and there is little to recommend it. A better bet would be to take Louisiana route 1 out of New Roads (near Baton Rouge) up to Marksville and then to Alexandria. This is much more scenic and the tracks are in sight much of the way.
As far as trains.....I haven't lived down there for several years, but there used to be about two trains each way a day between Baton Rouge and Shrewsbury yard in New Orleans. Train 9 & 10 (now I KCNO and I NOKC, I believe) are the primary premier trains, handling intermodal, but other traffic as well between NO and Deramus Yard in Shreveport. The other train was 55 and 56, which I believe are now symboled M SHNO and M NOSH (or something similar). There is a local which works the refinery and chemical plants at Norco and there are a couple of switch jobs at New Orleans. There is much more activity North of Baton Rouge along Route 61 in the maze of petrochemical plants. There are many local jobs and they are easily visible from the highway the entire length up the old L&A mainline to Port Hudson. These are all switch/local jobs, but at least you are guaranteed of getting a KCS train in motion. This is not the current KCS main which goes west across the Mississippi River Bridge (completed in 1940) then uses T&P (UP) trackage rights to gain its own rails.
I have driven between New Orleans and Alexandria many times while working in that area in the late 90's and early 2000s and never saw a mainline KCS train, so good luck! There is a little more traffic North of Baton Rouge, but it is very sparse between Baton Rouge and New Orleans.