In standard SQL, you use two single quotes to indicate one single quote, hence:
INSERT INTO SingleColumn(SingleChar) VALUES('''');
The first quote opens the string; the second and third are a single quote; and the fourth terminates the string. In MySQL, you may also be able to use a backslash instead:
INSERT INTO SingleColumn(SingleChar) VALUES('\'');
So, in your example, one or both of these should work:
INSERT INTO UnidentifiedTable
VALUES('Kellog''s', 'Corn Flakes 170g', '$ 15.90', '$ 15.90', '$ 14.10', '--');
INSERT INTO UnidentifiedTable
VALUES('Kellog\'s', 'Corn Flakes 170g', '$ 15.90', '$ 15.90', '$ 14.10', '--');
In PHP, there is a function to sanitize user data (mysql_real_escape_string) before you embed it into an SQL statement -- or you should use placeholders. Note that if you do not sanitize your data, you expose yourself to SQL Injection attacks.