The fact that Starr is black plays a defining role in her whole life—not just in her relationship with Chris. The fact that she didn't initially tell Chris that she was a witness to her friend Khalil's death, thereby creating distance between her and Chris, was due to her fear that he wouldn't understand the challenges that herself, Khalil, and other black kids in their community faced.
Khalil's death, at the hands of a brutal, racist policeman dubbed "One-Fifteen" based on the number that Starr could see on his uniform, sent the community of Oakland, California into turmoil in ways that could never have been predicted.
In reference to Starr and Chris's relationship, it is when Starr decides to go public and give an anonymous interview on television about Khalil's death that Chris figures out what's been going on. He recognizes her voice and asks her about it. In response, she confesses that she was indeed the person speaking, and gives him some insight into what it was like growing up in the projects. This proves not to be a deal-breaker for their relationship, and Starr and Chris confess their love for one another.
Simply put, the challenge that Starr and Chris face is two-fold. On the one hand, they are unfortunately forced to deal with the fact that due to having different skin colors, there are times in life when they will be treated differently. Second, they have to work through the fact that they have had vastly different experiences growing up. Many of these differences were caused by their different races.