The film plays most effectively — which is not quite to say that it plays effectively — as a story of two couples, Lee and Marina (a Russian-speaking Michelle Trachtenberg), drifting part, and Jack and Jackie, coming together. As the less well-documented, less fabulous pair, there is more room to move with the Oswalds, for the writer and the actors to make them their own. (Trachtenberg has the least-thankless job here; most viewers will have no conception of Marina to measure her against.)
Oswald became aware of Kennedy as it became clear the president was anti-communist. This offended Oswald who believed that communism was the way all political systems should operate. This belief did not change when Oswald returned to America.