Nowadays drawing a vector character or a portrait is pretty easy, except the eye drawing part. For vectorizing the body of the character you can easily trace the object of vectorizing with "Live Trace" (Object>LiveTrace>Tracing Options) in Illustrator or using the Artistic "Cutout" filter in Adobe Photoshop. Eyes are very important for expressing emotions. For creating live looking characters illustrators have to manipulate or draw detailed eyes. Here you can take a look at a basic tutorial for drawing vector eyes in Adobe illustrator, once you got the basics everything is improvisation. Enjoy !