Really? Not only are we given a clear idea of Burnham's personality in the first few episodes, we actually see that personality change in response to her situation within the first few episodes.
When we first see her, Burnham is supremely confident, a bit arrogant, but with a subdued sense of humor. In a flashback to when she first arrived on the Shenzhou, we see that Burnham's personality has evolved from arrogance almost to the point of contempt for her own species.
After the mutiny and her conviction, we see Burnham's personality change to sulleness, her confidence gone, She becomes tentative. Her personality has changed to reflect the guilt, humiliation, regret, fear (about her future).
Finally, after a series of events, she realizes she is still one of the best officers in Starfleet which brings much of her old personality back. That's partly on display when she "dares" to lecture Starfleet brass on the error of their ways.
Okay, that's just Burnham. I could go on because Tilly, Saru, Lorca, Stammets, and Ash's personalities are just as well defined.
How did you miss all this?