Is the phrase, "bawl my eyes out," appropriate for a personal narrative about events that occured in my life, or would it be considered slang and informal?
I think that much of this is going to depend on the requirements of the assignment and how much flexibility you will be given in appropriating informal language in your writing. Different instructors have different standards. Talking this over with your instructor is going to be a part of this process and will give you more insight on it. I do think that the the phrase is informal enough to be considered slang. That does not automatically disqualify it from being used in a personal narrative. It could be reflective of a moment in your narrative that needs to be illuminated and inserting a informal touch could be a way to broaden connections with the reader about an event that is really important to the narrative. That being said, there are other ways to phrase this experience. For example, using the simple idea of "I cried uncontrollably" is as effective as "bawl my eyes out." Along these lines, "sobbed mournfully" or "intensely wept" are other ways of phrasing it. If you really wanted to play with the idea, something along the lines of, "I cried like I never had before, tears that were connected to a universal condition of sadness and remorse were plain for all to see." Again, you might be able to simply rephrase the experience of crying heavily to avoid the slang issue in the first place.