From NYC to Charlotte is an 11 hour drive.
From Charlotte to Shreveport is about an 11 hour drive
Shreveport to NOLA is probably 4 maybe.
Charlotte to Nashville is an 11 hour drive
I'd guess Nashville to NOLA is another 11 hours.
What do you have in mind to do other than drive in 5 days.

The most direct route to New Orleans from NYC is to take the Pennsylvania Turnpike (I-78) to I-81 to its terminus at I-40, drive through Knoxville, turn south on I-75 to Chattanooga, take I-24 into northern Georgia and Alabama, then take I-59 all the way to New Orleans. You do NOT go near Charlotte and don't even come close to Shreveport (350 miles from NOLA, about 5+ hours, NOT four), which is on the opposite end of the state from New Orleans. The I-59 drive is crap - dull in Alabama (even if you want to stop in Birmingham); dull, lifeless, swampy and in the middle of nowhere once you cross into Mississippi.

You can take I-40 across Tennessee to see Nashville and Memphis and then go south on I-55 straight to NOLA or make a detour to Natchez and the plantation homes on the river. You'll likely need a stopover before Nashville, and there are far worse places than Lexington, VA.

Gretchen needs to check a map: Charlotte to Nashville is about 6:20 and I've done that trip; Charlotte to Shreveport is 843 miles, which is closer to 13 hours than 11; Nashville is 530+ miles from New Orleans, or about 8 hours.

Thanks so much Big Russ. I was doing it from memory, since I've done it all. It is 4 hours from Charlotte to Knoxville, and I believe it's a bit more than 2 hours to Nashville from there. They are long day's drives--not easy open roads (although all interstate) like out west where I can easily drive 12 or more hours.
My point for the OP, who can take whatever route he wants is that the distances themselves are full day drives for 3 days.

It is 20 hours of driving time according to Google, plus road time to get gas, eat, and use the toilet. So that is two 12 hour days with a night in between, a bit more if the weather forces you to take the coastal route.

Or you could do five shorter days and stop in Washington, Asheville, Nashville, Memphis, and maybe Clarksdale, Mississippi if you like blues.

This assumes that any time you might want to spend in NO is not part of your five days.

The weather this time of year is a real toss of the dice: you can hit snow anywhere on the route and ice over much of it. Ice is much worse because you can't see it until you are sliding backwards down the highway. Into a ditch. Ask me how I know this. The worst times are early in the morning and after sundown, and you will need to drive at these times if you are planning on the longer day option.

Would you consider riding the Amtrak Crescent? It leaves NYP at 2:15PM and arrives (if on schedule) 30 hours and 17 minutes later in New Orleans. This avoids at least 2 nights in a hotel room and a lot of driving.
When you have spent a day and a half in New Orleans, get on the "City of New Orleans" to Memphis. Spend 24 hours in Memphis and then go north on the CONO to Chicago. You get to spend close to 12 hours in Chicago before the Lake Shore Limited leaves at 9:30PM for New York. The fare in coach for this circle is $436 or less if you get a discount (AAA or Senior). More than twice as many people sleep in coach as sleep in the Roomettes and bedrooms.

Would avoid Amtrak like the plague. Amazed that anyone would recommend it for anything outside the Northeast Corridor: always late, always slow, and the OP specifically said the plan was to DRIVE, which means delays and stops are far more under your control.

If this is supposed to be round trip to NOLA and back to NYC in 5 days by car, however, it's completely daft - the drive time to NOLA from NYC is nearly 21 hours. I've done the trip from Northern Virginia more than once driving through the night and it's a haul.

Yes Amtrak is slow. The on-time performance for the Crescent is 81% for the past 12 months. http://www.amtrak.com/crescent-train&mode=perf&overrideDefaultTemplate=OTPPageVerticalRouteOverview
Over the past 11 years, I have traveled over 15K miles on Amtrak always in coach. The latest train I was ever on was the ViaRail Canadian (16 hours late to Winnipeg).
When you consider that you are riding toward your destination through the night instead of paying for a hotel room it is not such a bad idea IMHO.
Yes, I have driven over 1000 miles in 17 hours on a couple of occasions, but I choose to let others transport me long distance now.

The plan is a 5 day drive/trip from NYC to NOLA. After we get to NO we have about a week in NOLA which includes the superbowl and mardi gras, so the 5 day trip doesn't include NOLA-time.
According to Google maps, this is a 20 hours' drive, give or take, so that should give us plenty of time for doing stuff besides driving.
What are your recommendations for things to do in Nashville and Ashville? Also, I haven't heard of Natchez, what's there?

I am not from the US, so I do not know a lot about those places.

Thanks a lot!

Natchez is plantation life in the Olde South.

Nashville is a huge music scene, historically based on country music but more recently much broader. Lots of music in bars, yesterday's stars and tomorrow's. What Memphis and Clarksdale are to the blues and early rock, Nashville is to country and country rock.

Asheville is beautifully located in the mountains and is a budding hipster scene with lots of imaginative places to eat and a biggish (considering its size) art scene. Somewhat self congratulatory. For something completely different, outside Asheville is the Biltmore Estate, built by a Vanderbilt in the 1890's and owned by his descendants today. Modeled after a French Renaissance chateau, it is the largest private house in the United States (176,000 square feet or roughly 2000 m2) and has all the appurtenances one would expect in a grand demesne. Also the home of American forestry if your interests should tend in that direction.