Sir Raleigh has two attacks in this boss battle; one is a bounce attack where he bloats up and bounces around a platform or several platforms trying to smash Sly. The second is where he sticks out his tongue and spins around, trying to knock Sly off of the platform into the water.
When Raleigh first tries to attack Sly, you can either run around the platform you are on, or jump from one platform to the next to avoid the attack. Raleigh will bounce a total of five times until he shrinks back down to normal size. He will belch on the fourth bounce to let you know he is about to shrink down. Smack him with to take away one fourth of his health bar and start the next cycle. If you miss your chance at attacking him, or you are smashed while Raleigh bounces around, the cycle will start over.
The second cycle is similar to the first, but all of the platforms but the one you are on will sink into the water, making jumping from platform to platform impossible. Do the same as before, and attack Raleigh after his fifth bounce.
The third cycle is similar to the previous two, but the platforms will sink after Raleigh bounces on them. An easy way to complete this cycle is to jump around the arena, and when you land on the fifth platform, wait for Raleigh. He will shrink down and you can attack him.
The fourth and final cycle is different from the others. All of the platforms will sink, and Raleigh will land in the center of the platform you are on. Then, after a short delay, he will begin spinning in a circle with his tongue outstretched. You can tackle this part in one of two ways. One way is if you have a horseshoe (or you are replaying this level with the water safety technique), you can jump into the water and wait out the spin, then attack when you are bounced back onto the platform. The other way is to simply just jump over Raleigh's tongue as he spins. A single jump is best, as a double jump can cause you to fall into a second tongue spin. Raleigh will spin around a total of six times. After smacking Raleigh the fourth time, you beat him and the entire episode.