Develop your character in a small demo book or prepare some finished pages. More than just a design of a character, you should have a well thought out concept and even a finished issue ready to show publishers. This should be free of any errors in spelling or unfinished lines.
Make copies of your work using a high quality color or black and white printer. Never send originals to comic book publishers or agents. It is very unlikely that they will send them back, even though some places suggest sending a self-addressed stamped envelope so that your work can be delivered back to you.
Contact a comic book agent. Networking at comic book conventions or art meet ups is a smart way to get to know people who can help with your career. Follow agency submission guidelines in the hopes of finding someone to represent your comic.
Send your work to independent publishers. Again, follow each company's specific submission guidelines. Dream big, but think rationally -- it is extraordinarily unlikely that you will have your work read from Marvel or DC with an unsolicited piece. If you were unable to find an agency to represent you, find a small publisher to help get your work printed.