There's hardly room for more than a few marks when you're drawing an emoticon face. Believe me, you won't have room for the Sistine Chapel in 15 x 15 pixels.
You may like to print a few copies of the graph paper grid below for a warm-up scribble. Enlarge the grid some more before printing if you like.
Draw your facial expression first, then fill in the squares behind the lines to create a mosaic.
Each square on the grid represents only one pixel on a computer, tablet or phone display. There's no room for lots of busy detail. Even if you triumphantly manage to add some, as I did to Spotty here - - you probably won't be able to see them.
I actually made Spotty the same size as a basic favicon icon, 16 x 16 pixels.
Why is 15 x 15 pixels better than 16 x 16 for an emoticon?
At this impoverished size, a nice round number does not make a nice round face. You've probably noticed your face has a central protruberance known as a nose. Every pixel in a micro-portrait counts. If you draw an emoticon 16 pixels wide, even a two pixel nose can look enormous. Reduce it to one pixel and it unbalances the other features. One side of the face must be wider than the other.
It's easier to draw an emoticon on a computer
than it is to draw it on paper
Open Photoshop, Photoshop Elements or whatever art editor you happen to have. Most art editors have similar sets of basic tools.
Create a new canvas 15 pixels square.
Zoom up your image several hundred times so you can see what you are doing.
Set up a one-pixel pencil or pen.
Draw your emoticon face.
Drawing on a computer, you can work at a comfortable scale, while repeatedly zooming down to actual size (100%) to monitor how your emoticon is shaping up for its debut in real life.
Drawing an emoticon face directly on your computer has another advantage over paper. It's easy to erase expression lines that don't look right, while keeping the ones that do. Sad, happy, puzzled, crazy - whatever emotion you are aiming to express, you can chisel away at your image until you get the expression exactly right.