Answer by taking a quote from the following article:

Public Enemy is an American hip hop group consisting of Chuck D, Flavor Flav, Professor Griff, Khari Wynn, DJ Lord, and the S1W group. Founding member DJ Terminator X left the group in 1999. Formed on Long Island, New York, in 1986, they are known for their politically charged music and criticism of the American media, with an active interest in the frustrations and concerns of the African American community. Their first four albums during the late 1980s and early 1990s were all certified either gold or platinum and were, according to music critic Robert Hilburn in 1998, "the most acclaimed body of work ever by a hip hop act".

Carlton Ridenhour (Chuck D) and William Drayton (Flavor Flav) met at Long Island's Adelphi University in the mid-1980s. Developing his talents as an MC with Flav while delivering furniture for his father's business, Chuck D and Spectrum City, as the group was called, released the record "Check Out the Radio", backed by "Lies", a social commentary--both of which would influence RUSH Productions' Run-D.M.C. and Beastie Boys. Chuck D put out a tape to promote WBAU (the radio station where he was working at the time) and to fend off a local MC who wanted to battle him. He called the tape Public Enemy #1 because he felt like he was being persecuted by people in the local scene. This was the first reference to the notion of a public enemy in any of Chuck D's songs. The single was created by Chuck D with a contribution by Flavor Flav, though this was before the group Public Enemy was officially assembled. Around 1986, Bill Stephney, the former Program Director at WBAU, was approached by Ali Hafezi and offered a position with the label. Stephney accepted, and his first assignment was to help fledgling producer Rick Rubin sign Chuck D, whose song "Public Enemy Number One" Rubin had heard from Andre "Doctor Dre" Brown.  According to the book The History of Rap Music by Cookie Lommel, "Stephney thought it was time to mesh the hard-hitting style of Run DMC with politics that addressed black youth. Chuck recruited Spectrum City, which included Hank Shocklee, his brother Keith Shocklee, and Eric "Vietnam" Sadler, collectively known as the Bomb Squad, to be his production team and added another Spectrum City partner, Professor Griff, to become the group's Minister of Information. With the addition of Flavor Flav and another local mobile DJ named Terminator X, the group Public Enemy was born." According to Chuck, The S1W, which stands for Security of the First World, "represents that the black man can be just as intelligent as he is strong. It stands for the fact that we're not third-world people, we're first-world people; we're the original people." Hank Shocklee came up with the name Public Enemy based on "underdog love and their developing politics" and the idea from Def Jam staffer Bill Stephney following the Howard Beach racial incident, Bernhard Goetz, and the death of Michael Stewart: "The Black man is definitely the public enemy."  Public Enemy started out as opening act for the Beastie Boys during the latter's Licensed to Ill popularity, and in 1987 released their debut album Yo! Bum Rush the Show.

What else is interesting about Public Enemy in 1986 & 1987?
He called the tape Public Enemy #1 because he felt like he was being persecuted by people in the local scene.