This is so random but I had a thought so here we go.
"People worship me at Westerburg and I'm only a Junior!" <- Heather Chandler is in eleventh grade.
"I'll have to send my S.A.T scores to (whatever college she says)!" <- Veronica is in either 11th or 12th grade as only those years take SATs in America. (< Might be wrong, I asked google, I'm not American.)
"That's Jason Dean, he's in my A.P history class." <- A.P classes are taken only in years 10, 11, and 12.
Mac and J.D are both smart students (despite Mac referencing later in the film that she's flunking maths, but that's a totally different thing.)
None of this is concrete proof, or really narrows it down (aside from the first point, so here's some more.)
Ignoring these little bits of information, going by the real ages of the cast you could say:
Kim Walker: 18 y/o - 12th grade - movie canon: 11th grade (Around 16/17 years old)
Lisanne Falk: Undecided, actor was 23 when filming, probably 12th grade. (Around 17/18 years old)
Christian Slater: 18 y/o - 12th grade. (Around 17/18.) BTW, you have to be at least 16 for a motorbike licence in Ohio.
Shannen Doherty: 17 y/o - 11th or 12th grade.
Winona Ryder: 16 y/o - 11th grade
Bonus: Martha is presumably in Duke's year, Betty Finn is presumably in Veronica's year, Kurt and Ram are seniors (17/18, 12th graders).
Bonus Bonus: "Don't you have the Prom coming up?" - Some high schools in America only allow their seniors to have a prom, so Veronica is likely in the same year as Kurt, Ram, Martha, Betty Finn, Heather Macnamara, Heather Duke, and J.D, making Heather Chandler the youngest by a year out of all of the main students in the film.
All of this is for fun, and should be taken with a grain of salt!