To draw anime eyes on the computer, you can use a program like Draw With OekaManga Studio. If you're used to drawing them by hand, however, you'll have to learn a new approach, using the tools that are available to you on the computer. Here's an overview, along with a video tutorial.
Sketch 2 lines in which the position is similar to the letter “V”.
Draw the upper part of the eyes.
Draw the lower line of the eyes.
Add an oblong or a circle inside the eye area.
Add more details to the eye such as the pupil as shown in the picture.
Draw the eyebrow of the same length as the upper line of the eyes.
Add additional details like blush.
Erase the draft lines or hide the draft layers.
Set up your layers. If you do not know what layers are, it is a tool found on many illustrative programs. They can help make outlining and drawing on the computer easier. Set up about 4 layers (you can add more later).
Sketch out your eyes on the first layer. There are different types of anime eyes, however, they all have a shape to them that makes it identified as anime eyes. Search for anime images so you can get an example of what they look like, and use it as a reference. Use a fairly light color to sketch, as you are probably going to outline it in a dark color, and it will contrast. Your sketch does not have to be neat, just neat enough so you can trace over it.
Go to layer 4, (the first, second and third will be saved for coloring) and trace over your sketch. Use black to outline your eye, making it as neat as possible. You can choose to use the bezier tool, which would give you clean curves, freehand it, or both.
Use the eraser tool to erase the sketch on layer 1. If you're using the layers properly, only the sketch will erase, and not the black outline.
Use layer 2 to color in the skin surrounding the eye. Then, use layer 3 to color in the white part of the eye. Use a slightly blue gray to color the shadow the top eyelid casts down.
Color in the iris in layer 3. First use a base color, and then use a darker color around the edge and in the shadow; the top right/left,(or wherever the direction the light is coming from, see example). You can choose to blend it in, or not. Then, take a lighter version or the base color and apply it diagonally from the shadow.
You can choose to use the dodge tool to accent the eye, but don't overuse it.Draw your pupil using a dark color, if you did not include it into your outline. Then, add your glint to the eye by using a very light color close to white where the shadow is. Congrats, you're finished!