If you are an artist, you know how difficult it is to draw a 'cartoon me' avatar face by just looking at a photograph. Making the drawing look accurate while still resembling the person is the biggest challenge. There are very few artists who are able to draw caricatures very well.
It takes years of practice and a lot of tedious work. You need to draw every detail meticulously and then match all of the colors correctly. If your colors are too light or dark, this can have a definite impact on the quality of your drawing.
One of the newest ways that takes away a lot of the busy work of drawing a cartoon me avatar face by hand is to use several computer programs. These programs actually help guide you and draw all of the intimate details of a persons face. 
When you draw you have to be sure to make the caricature look accurate and funny, while not offending someone at the same time. This can certainly be an issue if you are trying to draw someone of another race or ethnicity.
If you ever are stuck and frustrated drawing a 'cartoon me' face avatar, you should sign up for an art class at a local community college. They have instructors who will help develop and hone your skill. However, there is no guarantee you will get results because most artists are just naturally talented. Some people try to draw caricatures for years and are never really able to make them look accurate and funny.