If you are specifying that it's only done today, you should put that into the sentence.
You could say one of the following:
I finished my class for today.
I finished today's class.
I'm done with my class for today.
I'm done with today's class.
However, most people will understand this by context.
The following conversation could be heard at the end of the day:
"Are your classes done?"
"Yes, I just finished them."
If it's still the middle of a term, it's assumed that the conversation is talking about their daily classes.
But if the same conversation were heard at the end of the term, when final exams are being written and classes are actually wrapping up, then it would be assumed to be not about daily classes but about them being done for good.