Kirk, a 31-year-old podcast-radio commentator and ally of US President Donald Trump, is credited with helping build the Republican Party's support among younger voters.
He was killed on Wednesday by a single gunshot in what Utah Governor Spencer Cox called a political assassination.
FBI and state officials said the killer arrived on the campus a few minutes before the event began, a talk by Kirk titled "Prove Me Wrong" outdoors in front of about 3000 people at Utah Valley University in Orem, Utah, about 65km south of Salt Lake City.
Security-camera videos show an armed individual ascending stairwells to get onto a roof before firing at Kirk, the officials told a news conference.Â
Kirk, a staunch defender of gun rights, was answering an audience question about mass shootings when the bullet struck his neck, prompting audience members to flee in panic.
The shooter jumped off the roof and fled into an adjoining neighbourhood, Robert Bohls, the FBI special agent in charge, told reporters.
Investigators found a "high-powered bolt-action" rifle in a nearby wooded area, and were examining that along with palm prints and footprints for clues.
On Thursday, with classes cancelled, the roof of the building on the otherwise deserted campus and the nearby woods were strung with yellow tape as investigators scoured them for evidence.
The shooter appears to be of college age and "blended in well" on the campus, Utah Public Safety Commissioner Beau Mason told reporters.
The shooter has not been publicly identified but the FBI circulated grainy images apparently taken from security cameras showing a "person of interest": a person wearing a black top, black sunglasses and a dark baseball cap.
"The FBI is offering a reward of up to $US100,000 for information leading to the identification and arrest of the individual(s) responsible for the murder of Charlie Kirk," the bureau said.
The shooter may have engraved messages related to transgender and anti-fascist ideology on their ammunition, the Wall Street Journal reported, but people familiar with the investigation told Reuters they were still being analysed and it was too soon to draw conclusions.
Trump said on Thursday he would posthumously award Kirk with the Presidential Medal of Freedom at a later date.
The United States' highest civilian honour, the Presidential Medal of Freedom is awarded by presidents to people who have made significant contributions in a variety of areas including culture and public service.
with DPA and AP
