Alex Olmedo, a Peruvian-born tennis star who won Wimbledon and the Australian Championships in 1959, becoming the first Latin American man to win a Grand Slam singles title, let alone two in a single year, died Dec. 9 at a hospital in Santa Monica, Calif. He was 84.

Mr. Olmedo died three days after another Hall of Famer and University of Southern California tennis star, Dennis Ralston. The cause was brain cancer, said his son, Alex Jr.

Known as “Cacique” or “Chief” because of his Inca heritage, Mr. Olmedo was a 5-foot-10 right-hander who lit up the amateur tennis scene in the late 1950s, winning top tournaments nearly a decade before Grand Slam events were opened to professionals. Official, computerized tennis rankings were still years away, but at his peak in 1959 he was considered the top amateur in the United States and No. 2 in the world.

Rightly or not, he was also deemed one of the sport’s most frustrating players, losing to second-rate opponents and struggling at times to elevate his game for matches he was expected to win. “There is nothing else quite like Olmedo in the formful world of tennis,” Sports Illustrated journalist James Murray wrote in a 1959 cover story. “He is the world’s best amateur tennis player one day. And he is plain run-of-the-mill the next.”

The son of a groundskeeper and tennis instructor, Mr. Olmedo grew up playing at a local club in Arequipa, Peru’s second-largest city. By most accounts, he came to the United States in 1954 after local supporters raised $700, enabling him to take a boat to Havana, a plane to Miami and a cross-country bus to Los Angeles. He spoke no English, and later spoke of a “guardian angel” watching over him on the Greyhound.

He went on to work at a tennis pro shop, where he said he “strung rackets and made malts and hamburgers” while developing his game, ultimately catching the eye of Perry T. Jones, the czar of Southern California’s amateur tennis scene. When Jones was named captain of America’s 1958 Davis Cup squad, he successfully lobbied for Mr. Olmedo — by then winning NCAA titles for USC — to join the team.

The decision irritated some current and former tennis players, as Mr. Olmedo was not American (he later became a naturalized citizen) and the tournament was a showcase for national pride. But Peru wasn’t fielding a team, and Mr. Olmedo silenced his critics after powering the United States to victory over Australia, ending a three-year winning streak by defeating Ashley Cooper, the reigning Wimbledon champion, in the finals on New Year’s Eve.

“We did it, Cap. We won,” Mr. Olmedo yelled to Jones, shedding a few tears on the court and throwing his racket in celebration. He received a congratulatory telegram from actor Kirk Douglas (“Now you are more than a chief. You are a superchief”) and was crowned with a laurel wreath during a triumphant return to Peru.

Mr. Olmedo won his first Grand Slam championship that same year, partnering with Davis Cup teammate Ham Richardson to win the men’s doubles title at the 1958 U.S. National Championships. Months later, he defeated Neale Fraser to win the 1959 Australian Championships, becoming only the second Latin American to win an individual Grand Slam title, after Chile’s Anita Lizana won the U.S. Nationals in 1937.

His victory set him up for the No. 1 seed at Wimbledon, where he defeated 20-year-old Rod Laver in straight sets in the finals of the world’s oldest tennis tournament. “He took his fast and aggressive game, and rocked the tennis world,” the International Tennis Hall of Fame later said.

Mr. Olmedo and Brazilian Maria Bueno, who won the women’s championship the next day, became the first South Americans to win individual titles at Wimbledon. He nearly won a third Grand Slam that year at the U.S. Nationals, where his former Australian opponent, Fraser, got revenge in the finals, winning in four sets.

But his success in major tournaments was accompanied by a number of disastrous matches. He was blamed for the American team’s loss in the 1959 Davis Cup, when he dropped two matches in the final round, and was charged with throwing a clay-court match after a lackadaisical performance near Chicago.

Sports Illustrated later reported that Mr. Olmedo had not wanted to play in the clay tournament anyway, and was insulted when he arrived at the club and the gatekeeper didn’t recognize him. In the predominantly White world of high-level tennis, he was sometimes mistaken for tournament staff.

“In Europe people treat you much better. I feel better over there because they know a player is a human being over there,” he said at the time. Exhausted by an amateur sports system in which he had to pay his own way to tournaments, then arrived to be treated poorly, he turned pro in December 1959, signing a contract that reportedly guaranteed him a minimum of $35,000.

Mr. Olmedo won the U.S. Pro Tennis Championships the next year and retired in 1965 with an overall record of 90-85. He was tired of traveling, he said, and wanted to focus on raising his family in Los Angeles, where he taught tennis until his death. He became the teaching pro at the Beverly Hills Hotel and worked with celebrities including Katharine Hepburn, Robert Duvall and Charlton Heston, describing himself as “kind of a therapist” on the tennis court, according to his son.

Greeting clients with a standard line — “How you doing, champ?” — he helped them with their serve or backhand while also chatting about life and work. “They come to see me to make themselves feel better about themselves,” he would say, “and to hit some tennis balls.”

The second of seven children, Luis Alejandro Olmedo y Rodríguez was born in Arequipa on March 24, 1936. His family said that Mr. Olmedo started playing with a homemade wooden racket by age 4 or 5, and within a decade had beaten everyone in town. He went to Lima, the capital, where the head of the Peruvian tennis association connected him with American coach Stanley Singer, setting him on a path to come to the United States.

Mr. Olmedo was 15 when he made his amateur debut at the 1951 U.S. Nationals, losing his first match. His son and former wife, Ann Peirce Olmedo, said that he had already made his epic boat-plane-and-bus trip to Los Angeles by then, in contrast to news reports that suggested he made the journey three years later.

By all accounts, he learned English at night school soon after arriving, and studied at Modesto Junior College before enrolling at USC to play under tennis coach George Toley. He won NCAA doubles and individual championships in both 1956 and 1958, and helped lead the Trojans to a 1958 team championship.

Mr. Olmedo was inducted into the International Tennis Hall of Fame in 1987.

His marriage ended in divorce. In addition to his son, Alex Olmedo Jr. of Santa Monica, survivors include two daughters, Amy Olmedo of Los Angeles and Angie Williams of Paia, Hawaii; four brothers; a sister; and four grandchildren.

In a phone interview, his son recalled that Mr. Olmedo was still leading private lessons until shortly before his death, working with clients on an extra-wide, extra-long court that Mr. Olmedo had installed outside his home in Encino.

“When he was around people, he would just turn on the charm,” Olmedo Jr. said. “If you were a beginner, he was so precise he could hit the ball exactly to your sweet spot and make you feel like you were a better player than you were. That’s the key to being a pro, making people feel good about their abilities while also helping them improve. People would come back and say, ‘What’s going on, how come I can’t play as well against other people as I do against you?’

“And he’d say, ‘Well, champ, you’ve got to keep coming back.’ ”