Although Franklin Roosevelt was elected to an unprecedented four terms, he had his critics. Who were these critics and what did they offer as an alternative?

Most of the criticism against Franklin Delano Roosevelt was aimed against his New Deal policies. The New Deal was a series of socially liberal legislative programs aimed at ending The Great Depression and creating protections to ensure that another depression would not occur. Some New Deal programs, such as the Works Progress Administration, were temporary and aimed at rekindling industry and business by employing men in public works projects. Others were intended to be permanent, such as the Federal Deposit Insurance Corporation (FDIC), which insures customers' accounts (up to $250,000) against bank defaults.
Father Charles Coughlin was one New Deal critic. Coughlin hosted a popular radio program. While, initially, he had been supportive of Roosevelt, his favor had faded by 1934. Coughlin thought that Roosevelt's initiatives were still too friendly to banks. He argued instead for the nationalization of certain institutions, such as the Federal Reserve. He was staunchly against private interests, arguing that financial institutions and industries ought to be publicly owned.
The Supreme Court, too, was an adversary. It deemed some of Roosevelt's proposals to be unconstitutional. One such program was the Agricultural Adjustment Act (AAA). They decided, in a majority vote of 6-3, that it was illegal to levy a tax on one group to compensate another. Roosevelt had wanted to tax food processors to compensate farmers. 
Some of Roosevelt's adversaries were political rivals. Huey P. Long, governor of Louisiana at the time, was one. He criticized The New Deal in similar ways as Coughlin had, alleging that some of the programs amounted to "corporate welfare." Also, like Coughlin, Long had been supportive of Roosevelt's first presidential campaign. However, he had ambitions to run for the presidency himself, which led to his adoption of an adversarial stance as well as the creation of his own solution to lifting the nation out of depression.
Long called his plan the "Share Our Wealth" plan, which amounted to a series of Socialist measures, primarily the enforcement of a standard of living. He intended to cap fortunes at five to eight million dollars ($60-$96 million today).
Joseph P. Kennedy was also a political rival. While Roosevelt planned to go to war with Hitler, he perceived that Kennedy stood in his way. Kennedy, while at home, favored isolationism. Though Roosevelt took Kennedy's political acumen seriously, he exploited his social awkwardness. Kennedy was, first and foremost, a social climber and wanted the kind of respect that Roosevelt -- a member of one of the nation's oldest and most respected families -- had long enjoyed. To appease him, and to get him out of the way, Roosevelt handed Kennedy an ambassadorship to England. 
Franklin D. Roosevelt had enemies on both the left and the right. While no one is surprised to learn of the Republican opposition against him, it is interesting that three Democrats -- Father Coughlin, Huey Long, and Joseph Kennedy -- were also enemies.
http://www.dtman.com/steve_private/school/newdeal.htm

https://www.hueylong.com/programs/share-our-wealth.php

https://www.newsweek.com/when-franklin-roosevelt-asked-joe-kennedy-drop-his-pants-283146