If we said "I am crying of happiness," which we don't, it would mean "I am crying about happiness." The English idioms are "these are tears of joy" or "I am crying tears of joy," depending on whether the listener already knows that you are crying. If you don't want to mention tears, "I am crying from happiness" or "I am crying out of happiness" would be ok. We do say "a cry of joy," "a cry of pain," and so forth, but in those contexts "a cry" means "a shout."
You could say "crying WITH happiness" or "crying with joy," but NEVER in English would someone say "crying of happiness." I put down "I am crying because of happiness" because at least, though awkward, it came somewhat close to what is meant, and isn't incorrect English. Your translation is poor English and no native English speaker would ever say it.