You are right in asserting that Ondaatje's work is complicated. It is. No doubt about it. You will have become very used the idea of narratives flowing into one another with different backgrounds that converge into reality. This is tough.
The fundamental context of all of the characters in the book is World War II. Different elements of this conflict present themselves to different characters. Yet, in the end, all of the primary characters find themselves struggling with the conditions of war and the impact it leaves on the human being. Hana is a Canadian nurse during the war, and one who struggles with death from both a universal and personal perspective. She is haunted by the constant presence of soldiers' deaths, and her inability to stop it. At the same time, she is haunted by her own father's death and the death of her unborn child via abortion. Kip is an explosives expert, working for the British army, yet frustrated at the British control of India and the Western colonization that has suppressed his country. Caravaggio is an Canadian spy of Italian descent, and a victim of German torture. The English patient might be the most complex of all of the characters here, though. He is considered English because of his grasp of language. Yet, his identity is not really established in terms of his affinity with political sides. He is a brilliant cartographer of the North African desert, who is in love with Katherine Clifton, the British wife of one of the Patient's colleagues. After the tragedy in the desert, he is unable to save Katherine because of his name. He can only go back after two years, to retrieve her body, and in doing so, he suffers the accident that disfigures him. All of these character narratives are woven with the background of the war, and expressing its pain and brutality becomes the primary focus of the novel.