Begin by making a square (hold shift key down to make a perfect square) …
Next, use the line tool to create a simple line.
Use the reshape tool to start adding points and shaping your line into the contour you want.
When you have something of interest, copy it and put the copy on the opposite side. Run your tile program to see you it looks when tiled. Starting to see an interesting shape? Keep working back and forth with identical lines on opposite sides until you have something you like.
Now move on to the other sides. Again, start with a simple line…
Add more points to make the shape interesting– I see a snout forming…
Check to see how the exact same line looks on the back side (make a copy of your line and move it to the back). Run your tile program to get a rough idea of how it looks. Once you’re happy with the rough you can tighten it up to tile smoothly.
I like what I have so I’m going to move the line segments in close to align them more perfectly. When I’m happy with them, I will fuse the four line segments into one new polygon.
I use the reshape tool to bring the end points of each line close to the end points of the neighboring lines. With a little wiggling at each corner the end points will snap together and connect the lines into one big shape. Do one corner at a time.
Now my four lines have become one continuous outline. Check to make sure it tiles nicely (you may need to move some of the points around a bit to make a smoothly tiling shape. Delete the golden square when I’m ready to start coloring my final shape.
Fill in with a basic color.
Make some circles for an eyeball.
Put in the eye!
A few subtle touches to make the character sparkle with life: add a white circle as a highlight on the pupil, and perhaps an ever so subtle gradient on the “white” of the eye.
Apply an overall gradient to the body with the Fill Tool in gradient mode.
I added a colored circle (no outline) and twisted the circle shape around with the reshape tool until it looked like a nice wet highlight on the body. 
Make some new polygon shapes that will fit in place as fins. Start each as a rectangle, then add points to turn your rectangles into fins.
Add some gradients…
Plop in your fins!
Add some line segments to give your fins some ribs. You can apply gradients to lines if you want to add that little bit of nuance.
Done!  Wow, if you did even a chunk of this then, seriously, you are now a bonafide vector illustrator and character designer. Congratulations!!