Cookson has been president of British Cycling since 1996, overseeing high profile Olympic and Tour de France victories that have encouraged many ordinary Britons to take up cycling to keep fit or commute to work.
Cycling globally is struggling to emerge from the shadow cast by Armstrong, the American who was stripped of his seven Tour de France titles last year for doping offences.
"The first priority for the new UCI president must be to change the way that anti-doping is managed so that people can have confidence in the sport," Cookson said.
Critics have accused the UCI of failing to do enough to catch Armstrong during the years when he dominated the sport.
"We must also urgently carry out a fully independent investigation into the allegations of corruption in this area which have so damaged the UCI's reputation," Cookson added in a statement announcing his plan to stand.
McQuaid, who took over as president of the UCI in 2005, the year of Armstrong's last Tour de France victory, has strongly defended the federation's handling of the Armstrong affair.
McQuaid plans to seek a third term as president in an election in the Italian city of Florence in September.
Cookson, who has served on the UCI's Management Committee since 2009, said he told McQuaid that he would challenge him in what sounded like an awkward phone conversation.
"I called him on Saturday and informed him that I was standing," Cookson said. "I think you can guess, he was not particularly happy but there you go."
Cookson said that big companies would be reluctant to invest heavily in cycling until they were convinced that their brands would not be tainted by doping, holding back the sport.
"This is very much an activity whose time has come," he said. "It's great sport, it's good for health, it's good for transport, it's good for the environment."
Cookson said he believed the sport was much cleaner than in the Armstrong era, but still had work to do.
"There has been a huge improvement but I think there are still people out there who are trying to cheat and we've seen evidence of that in the Giro this year," he said. Two Italian riders failed dope tests at last month's Giro d'Italia.
Jaimie Fuller, an Australian businessman who has led the "Change Cycling Now" lobby group, backed Cookson.
"We believe him to be a man of integrity. He has hit the nail on the head with the issue of doping," said Fuller, who is chairman of the SKINS sportswear company, a cycling sponsor.
"He's shown a track record in the UK that he can deliver," added Fuller, who has repeatedly called for McQuaid to quit.
"Change Cycling Now" had previously proposed American Greg LeMond, who won the Tour three times, as an interim head of the UCI.