Translation:This girl has a very good personality.
"A good character" might sound old-fashioned but still has currency. Depending on what you compare "character" with you could find paralells with or without "a"; I'd think both should be accepted.
a good temperment/ a good personality/ a good conscience
good willpower/ good resolve/ good judgment
Using "a" wouldn't necessarily mean 'character role', whether she is an actress or playing an RPG or anything else.
(American English speaker, to frame the below) "She has a good character" strikes me as being the only way of expressing a very particular meaning: her "character" (i.e. her personality, temperament, etc, in addition to just her moral mettle) is a good one (whatever one might mean by "good" here, which could depend on personal taste in acquaintances). "She has good character" strikes me as being more (even fully?) focused on her (significant) moral fortitude.
It's worth repeating what Dmitry_Arch says below. The Russian sentence is about the girl having a nice personality. "характер" and "character" definitely have a dollop of false friendness about them. I recall an exercise in Russian class where we had to tell what our "характеры" were like. Potential answers included things like "sociable," "good sense of humor" that probably wouldn't be the first things to spring to mind if somebody asked you, "What is your character like?"
Speaking of a person, добрый means kind/good-natured. хороший человек means "a good person", so it is more general. For a good deed (an act of kindness) both доброе дело and хорошее дело are used interchangeably. The same applies to добрый/хороший совет (good advice). The word добрый is older than хороший and is found in all Slavic languages (хороший is only found in Russian and Ukranian), Using добрый instead of хороший to describe inanimate objects sounds slangy in modern Russian. Here are some set expressions with добрый: Доброе утро!, Добрый день!, Добрый вечер!, Доброй вам ночи! Желаем Вам доброго здоровья!, В добрый путь! (Farewell!), добрый мОлодец (an honest youth), по доброй воле (voluntarily, on one's own free will), добрый гений (genius loci), добрый нрав (good temper). No, good temper = спокойный нрав, добрый нрав = good nature
dobre is used more for other slavic languages like Polish or Czech, Russians wanted to be unique but their country was in a very hectic state at the time. We may go as far as to say, it was a Horror-Show. They didn't mind it though, and so they said все хорошо. Getting back to seriousness though, If you speak Polish its like the difference between dobre, and dobrze. Both mean good, but you use them differently. dobry dzień- description word for something else. Dobrze- it's good, Okay, Affirmative, etc. The problem is that as time passed russian has also began using it in the aforementioned way, which causes confusion to foreigners. Pretty much, whenever it's declined through a case, it can be used differently.. This may have hurt more than it helped..
The Russian sentence implies that the girl is good-natured, kind, caring, pleasant to deal with. She wouldn't complain about small problems, nor would she lose temper over trifles. On the other hand, if you describe someone by saying, "он/она с характером", it will mean that the person is stubborn and tends to do things their way. It also implies that the person has strong personality. The idiom is used in a positive way - usually when you are a little surprised.
In English, saying that someone has a good temper generally means that they are emotionally stable, that they won't get angry or aggressive. Good character implies honesty and integrity. Some with character is probably independent-minded and willing to stand up for what they believe is right, even if it is inconvenient.