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His origin story is a lot different. First off, in the comics he was born in W'kanda, not Oakland. Second, he's not related to T'challa at all.
In the comics, Klaw basically forced him to work with him, and when T'challa forced Claw out of Wakanda, Klaw took Killmonger with him to America, where he then escaped.
In the comics, Killmonger wasn't in the military at all, in fact he got an education and even got a PhD in Engineering from MIT.
As for the rest, it's fairly accurate. They omitted some extra details like Killmonger getting the help from another guy in raising a zombie army, and getting saved in the end by a boy who ends up killing Killmonger himself.