Richard's a bit of an outsider with something to prove, so while he hangs out with the aristocracy, he starts the play technically outside of their elite circle. This is because his father was beheaded for treason, which means his son can't just inherit the title. Bummer for Richard.
Richard's also feuding with Somerset about something, and the argument grows into a tension between their two families, symbolized by white roses (York) and red roses (Somerset). This is ominous enough in its own right—things get legit thorny (ha), and Talbot even ultimately dies as a result of their squabbling—but it would have been even more foreboding for Shakespeare's audiences at the time who knew the families would soon be fighting an all out civil war.
But the civil war doesn't start in this play. What does happen is that Richard is restored to his family title of Duke of York—yay—and he also discovers he may have a better claim to the throne than Henry VI. Hmmm… Wonder what he's thinking about that? We get hints that he may want the throne, but he doesn't go very far along that road in this play. Just wait until Henry VI, Part 2, though.
In this, however, we do see that Richard's not afraid of a fight. He keeps arguing with Somerset, and he's pretty angry when the English make peace; he's also the one who captures Joan of Arc, and he taunts her before she's burned at the stake. This establishes that York can be mean, but also that he's bold. York thinks he's not one to mess with—and he's totally right.