Train from Barcelona to Rome

There is no direct train or night train on that route; hence your topic is actually taking a series of "Trains from Barcelona to Rome."
If you are not planning on stopping off over night in some towns like Nimes (there is high speed rail to Nimes from Barcelona) or Nice or Monterosso on the Italian side along the way between Barcelona and Rome you should just fly. It will be a lot less expensive as well. The trains are often running very behind schedule on the Italian side, which would cause you to miss connections wasting your time.

I have taken the train from Barcelona to Milan. It wasn't has scenic and relaxing as I thought it would be. As Jazz said, you will be changing trains. In your case-Nimes, again at the border with Italy, probably again in Genoa, and depending on route from there possibly again. So no, not a good trip in one stretch or ticket.
I stopped in Nimes for two nights to check out that area. By breaking it up, it was just 3 hour blocks on the train. As far a scenery, along the coast it seemed to be in a lot of tunnels and only small bits of cool views.
There will be no sleepers available since you will be changing trains. I would only take that long of train ride if I could break it up with overnight stops along the way.

Thanks so much. Doesn't sound like a great idea. I really appreciate the help.

If you have a few days and want to make stops along the way, it can be a great idea!

Although there is no need to change train in Nîmes, but if you want to make a stop there it is certainly possible.