Know the market. Read a lot of cartoons, and research characters to gather an idea of what sort of material is out there. Make a list of publications that you would like to submit work to, read back issues and see what kind of cartoons they are buying.
Jot down some ideas for a storyline. Envision pictures to go with them, and illustrate your vision. Draw as many as possible to further develop the concept in your mind. When you reach up to 12, stop and focus on what you have accomplished.
Think about what sets your cartoon characters apart from what is currently on the market. Review your characters' facial features, style and concept. Make sure that they are publisher-friendly but have strong originality before proceeding with the project. Following a current trend can work; however, publishers seek uniqueness, as well.
Whether you are making single-panel style magazine, comic strip syndicate or greeting cards cartoons, decide what kind of style fits best with your cartoons.
Create a portfolio to showcase your work. Read magazine submission requirements and send materials.
Create a website to add more visibility to your cartoons and brand. It can generate enormous buzz, start a fan base and grab the attention of publishers scouting for new, undiscovered talent.