Such an extreme degree of emotional lability could indeed be a sign of an emerging Bipolar D/O, and the positive family history increases the chances of this. Also, emotional fragility can be a by-product of inadequate parenting/nurturing during the earliest years, even in the absence of any organized mental health condition. However, it is fair to say that anger and mistrust are more common symptoms than emotional fragility. Five-year-olds can be prescribed medication to treat mood disorder, but it's not something that's favored unless the symptoms are so serious that they pose a major impediment to a child's functioning. In the best light, what you're witnessing is more of a developmental problem, in particular the lack of capacity to manage frustration. If this is true, her ability will likely improve over time, and you can focus for now on helping her adopt a routine response when she's frustrated. For example, a systematic plan would be for her to either (a) tell you that something is bothering/annoying/frustrating her, or (b) cry, but go to a designated place - i.e., her room - to do so.
This is my daughter exactly! I adopted her when she was 2 (she's 6 now). I believe at first the crying was her way of communication as she didn't have the language skills to express her needs & frustrations (she used to have major tantrums too) but over time the crying mainly became a habit. She does it less at home now because she has learned that she doesn't get my attention from it--I just tell her to go to her room until she is done crying and that if she wants something from me she needs to use words to tell me. At school, however, her teachers tell me she cries frequently--but with some of the situations they have told me about, they are sometimes giving her more attention when she cries and even allowing her to get by with things the other kids aren't allowed to do in order to prevent her from crying. So at school she is still getting what she wants by crying. Good luck with your daughter!