Just once — or not even once if you consider North, Central and South America as three distinct geographical divisions. Columbus wanted to reach India by sailing west and his plan was financially supported by Queen Isabella of Spain though Columbus himself was an Italian by birth. He set sail from Spain in the ship named Santa Maria, reaching Bahamas on October 12, 1492 and later Cuba and Hispaniola (Haiti), which are not parts of North America. His second voyage during 1492-93 led to the discovery of several Caribbean islands. This is when he actually came nearest to setting foot on North American soil, because he stopped at Puerto Rico which today, is officially part of the USA. However, the connection is political, rather than geographical, because Puerto Rico lies in Central America. On his third voyage (1498-1500), Columbus (picture, above) discovered the South American mainland. This expedition took him to Venezuela and the mouth of the mighty Orinoco River. His last great voyage (1502-04) was along the south side of the Gulf of Mexico.
On none of these four historic voyages did Columbus visit North America as we know it today, though the issue ultimately rests on whether you make a distinction between Central and North or not. That apart, the fact remains that Columbus was truly the discoverer of the New World.