How do I improve the rating of my crops?

Nope, you're doing it correctly. It takes somewhere in the vicinity of a half dozen cultivations to increase their rank. You'll know it as the cultivation will succeed, and then you'll get a little celebration, showing that your crop has ranked up.

Incidentally, that's the same way you win contests. You can see the contest rank on the Actions screen, and if you actually have a crop in that rank, I've found that it's an almost guaranteed win. So entering a D rank crop into a D rank contest generally means you win, along with the benefits of a contest win, which is a small popularity boost, and a global 10/20+% increase in that crop's sale price. So winning several contests with the same crop can make it incredibly profitable.