It means you criticize them harshly and angrily. Probably not crying, but shouting possibly. In American English we would use the phrase "to rip into someone" or "chew someone out."

To "have a go" at something or someone means to make a concentrated effort to do some (unspecified) thing. It has no necessary connection with anger or anything nasty, though in certain contexts it does. It could also be something wonderful. For example, if you are watching people ride horses but have never ridden one yourself, you might someday decide that you will "have a go at it". That means you are ready to try doing it yourself. Suppose you are a teacher trying to explain mathematics to three students: Billy, Sally, and Tom. In talking with another teacher, you might say "Sally and Tom catch on to everything I say, but Billy never seems to understand". The other teacher says, "I am willing to try to help him. You teach Sally and Tom and I will have a go at Billy."

Hi

Actually it can also mean to try and fight them.

'to have a go at someone' could mean to criticise them/argue etc to their face, or could be done behind their back too in the form of criticism.