The U.S. Constitution says "Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors . . ." 
The U.S. Supreme Court has repeatedly characterized the authority of the state legislatures over the manner of awarding their electoral votes as "plenary" and "exclusive."

The normal way of changing the method of electing the President is not a federal constitutional amendment, but changes in state law. 

Historically, major changes in the method of electing the President have come about by state legislative action. For example, the people had no vote for President in most states in the nation's first election in 1789. However, now, as a result of changes in the state laws governing the appointment of presidential electors, the people have the right to vote for presidential electors in 100% of the states.