As I know it in the United States, the expression is always "don't count your chickens before they've hatched." It means "you are being overconfident. You are acting as if something were sure, when it is uncertain." "Do you think I should wear a blue dress or a green dress to the awards ceremony?" "Don't count your chickens before they've hatched! You haven't even entered the contest, and you might not win."

Since Charles doesn't know his marks yet (i.e. his grades), he can't be certain he will graduate. For example, he may have failed a course he needs to graduate. Nevertheless, he is already planning his graduation party because he assumes he will graduate. Thus, he is "counting his chickens before his eggs have hatched." Similarly, a farmer with 10 eggs can't assume all 10 eggs will hatch and produce 10 healthy baby chickens; a farmer that assumes that 10 eggs now guarantees him 10 chickens in the future is also "counting his chickens before his eggs have hatched." In Spanish, we might say something like "no cantes victoria (antes de tiempo)" or "una golondrina no hace verano." Hope this helps!

Counting your chickens before your eggs have hatched means that you are making plans assuming something that is not guaranteed will happen. In this expression, Mark is planning his party assuming he has the grades required to pass his classes, and in turn receive his degree. Therefore this idiom, "don't count your chickens before your eggs have hatched", is used to caution people not to take certain events for granted.

before his eggs

Hi Diego, other commenters have done a good job explaining the expression so I won't go over that again. I did want to let you know that people often shorten the expression to "don't count your chicks before they hatch". I would say it's more common to see that version that the full one you quoted.