His killer, Mark David Chapman, was actually a huge Beatles' fan & idolized Lennon at one point. But then two things happened that changed his perception of Lennon.
First, he became a "born-again Christian" so he started resenting some of Lennon's remarks regarding religion. Specially Lennon's remark in 1966 that "The Beatles were more popular that Jesus." This quote has ALWAYS been taken out of context & misinterpreted. Lennon WAS NOT implying that The Beatles were "better" or "more important" than Christ. He was condemning the fact that young people were not into religion anymore & that more kids knew about The Beatles & their lives than they did about Jesus & HIS, & that to kids, "The Beatles were more popular than Jesus."
Second, he read J.D. Salinger's "Catcher in the Rye", which is about a teen-age boy (Holden Caulfield) who becomes disillusioned with the world & thinks that adults are ALL hypocrites (saying one thing & doing something else). Mark David Chapman began seeing himself as Holden & decided to rid the world of hypocrites...in this case, John Lennon, who Chapman saw as a hypocrite because of his song "Imagine" where he sings, "Imagine no possessions." yet he owns mansions, etc.
So, because of these things the voices in his head told him he should kill John Lennon