So there is a cartoon character that captures exactly the spirit and character of you, your business or your website. Before using that character for any purpose other than personal or educational, you'll need to get permission to do so. Cartoon characters, like every other work of art or literature, are protected by copyright laws, meaning the original creator has the right to control how the work is used -- and, incidentally, to profit from the work, if they choose.
Find the name of the copyright owner. For a cartoon character, the syndicate or the publishing company likely holds the copyright for the artist. Even if the artist holds the copyright personally, the best way to contact the artist is through her publisher or syndicate.
Look up the copyright owner's address or email address. For large companies such as United Feature Syndicate or New York Times, there is a section of the company website explaining how to contact the copyright or intellectual property division. For smaller publishers or individual artists, use the general email or physical addresses.
Write a letter or email requesting permission to use the cartoon. In the letter, include the following information: Who you are, the purpose for which you would like to use the material, the exact image you would like to use, how many copies you will be making and how they will be distributed. For example, tell them if the image be used in a class, a training seminar, a PowerPoint presentation or on a website.
Wait for a response to your request. Some possible responses include granting permission for the limited use, asking you to pay for the use, and not granting permission.