The first fight of the World Circuit is against Irish character Aran Ryan. As Don Flamenco he has some tricks up his sleeves, like his power move a Missile Head butt. There is little chance dodging his basic attacks and to attack him directly is nearly impossible, but you can counter him a couple of times to give him some damage. His power move is more difficult to avoid and is not possible to block because he will hurt you if you try to block him. When he use his power move do a right jab to repel him from hurting you.
The second fight is against another classical character, Soda Popinksi. It's probably the fight that you will lose for the first couple of times. Soda is not difficult but, he's very quick. There is no real trick except to memorize his patterns and counter punch his jab to knock him down.
The third fight of the World Circuit is against the Bald Bull, his strategy changed a little bit but he is not difficult; try to avoid his Bull Charge or stop him by throwing a jab. Hit him in his jaw and try to win by knock out.
The fourth fight is with one of the most emblematic character of the Punch Out franchise, Super Macho Man and maybe the easiest fight in the entire game. Like the original counterpart he will use a series of uppercuts to try to decimate your energy bar, after you knock him for the first time he will use a combination of jabs and clotheslines. To put an end to the fight try to dodge them and repeat the process by hitting him to defeat him.
The last fight is with Mr. Sandman. He is a combination of Don Flamenco and Baron Von Kaiser where he has a set of quick and delayed moves just to stop you, but his basic combo is the least of your worries. His power move, the Dreamland Express, is similar to the power move of Don Flamenco but if you are hit by the power move you will be instantly defeated. He says "Dreamland" before he attacks you, but he will try to fool you by saying "Boo" a couple of times before executing his power move. Mr Sandman always throws the same pattern of punches, and you might lose a couple of times before defeating him because you have to memorize his pattern.
After you defeat Sandman you will see the credits and a supposed ending, but the game doesn't end there. There is a round two with the same characters in which you have to defend your title as the world champion.