Well, what kind of column is Column C? Does it have a default value? Depending upon these questions, you'd have a query that looks like:
UPDATE `myTable` SET `column_b` = `column_c`, `column_c` = ''
-- OR --
UPDATE `myTable` SET `column_b` = `column_c`, `column_c` = 0
-- OR --
UPDATE `myTable` SET `column_b` = `column_c`, `column_c` = DEFAULT