Cook was born in Akron, in Erie County, in western New York. He moved to Louisville at 17. Also at that age, he joined the US Navy and served on submarines in both the European and the Pacific Theaters of Operations during World War II. After the war, he enrolled at the University of Louisville and earned a Bachelor of Arts degree in 1948 and a law degree in 1950. He practiced law in Louisville until 1957.
Cook was elected to the Kentucky House of Representatives in 1957 and again in 1959. He served on a special committee analyzing education in the state and also on a planning committee.
Cook was elected to two terms as Jefferson County Judge, the equivalent of a mayoral or county executive position administering populous Jefferson County, which, by the 1960s, was mostly suburbs of Louisville. He was elected in 1961 and, along with fellow Republican William O. Cowger, who became the new mayor of Louisville, Cook unseated the Democratic Party, which had held both offices for 28 years.
In 1962, Cook was primarily responsible for the county's $34,000 purchase of the decrepit steamboat Avalon at public auction in Cincinnati, Ohio. Auctioned as little more than scrap material, upon refurbishment the boat was now called the Belle of Louisville, and, as of 2007, it still carried passengers yearly and was one of the most recognizable symbols of the city. At the time, Interstate 64 was being constructed along the city's waterfront, and Cook's purchase of the steamboat was intended as a measure to bring attention to the city's historic cobblestone wharf.
A politically-motivated taxpayer suit was brought by local lawyer Daniel Boone because of the county's expenditure of such an "outrageous sum" for a dilapidated "throwback to the Dark Ages of transportation," in Alan Bates' memorable phrase. According to Cook, the expenditure worked out to roughly six cents per taxpayer, a negligible sum, even at that time, and when individual citizens complained, he would simply pay them off with pennies from a jar that he kept in his office desk for the purpose. In a 1989 interview, Cook said that some people insisted on checks, and he wrote several such six-cent checks, none of them was ever cashed.
Cook was reelected county judge in 1965 by a wide margin, 121,481 votes to Democrat William B. Stansbury's 71,280.
In 1967, Cook ran at the top of a slate of statewide office holders as a candidate for governor of Kentucky in the Republican primary election. He was narrowly defeated by more conservative Barren County Judge Louie B. Nunn, who went on to be elected the first Republican governor in Kentucky since 1943. Nunn had also been the party's unsuccessful gubernatorial candidate in 1963 but had narrowly lost to Democrat Edward T. Breathitt. At the time, Kentucky governors could not succeed themselves in office.
In 1968, Cook ran for the US Senate to fill the vacancy created by the retirement of another moderate Republican, Thruston Ballard Morton, a former chairman of the Republican National Committee. In the general election in which Richard M. Nixon carried Kentucky over Hubert Humphrey and George Wallace, Cook defeated former state Commerce Commissioner Katherine Peden. He was the first Roman Catholic to hold statewide office in Kentucky. He was subsequently one of the first Republican senators to call for Nixon to resign during the Watergate Scandal.
Cook was defeated in his 1974 bid for re-election by Governor Wendell Ford, a popular Democrat. Cook's repeated plea that Ford debate him was seen as highly unusual. Following the election, Cook resigned his seat early, in December, so that Ford could resign and be appointed senator by his successor, thus having greater seniority in assuming the office. (Morton had done the same for Cook, in 1968.)
Following his political career, Cook practiced law in Washington D.C. until 1989, when he retired to Sarasota, Florida. In a fiery op-ed, he announced his support for Democrat John Kerry of Massachusetts in the 2004 presidential election: "I have been, and will continue to be, a Republican. But when we as a party send the wrong person to the White House George W. Bush, then it is our responsibility to send him home if our nation suffers as a result of his actions."
Some of his former aides went on to congressional careers. Mitch McConnell, now the Senate Majority Leader, was Cook's chief legislative aide from 1968 to 1970, and John Yarmuth, a representative, was an aide to Cook in the 1970s until the former became a Democrat.
Cook, however, opposed McConnell in the 1984 campaign. McConnell defeated the incumbent Democratic senator, Walter Huddleston.
In later years Cook, was uncertain about what he considered McConnell's turn to the right. McConnell had helped Cook to advance the unsuccessful Equal Rights Amendment, but Cook opposed his former aide on several other pieces of legislation.
Cook died in Sarasota from complications from a heart attack, at 89.