It's quite difficult to get something trademarked - in fact, something like half of all submitted trademark applications fail. I don't know whether most of those are filed by non-lawyers, but even lawyers are sometimes unable to get their applications accepted, so there's not much doubt that your chances of getting a trademark are much better if you consult a lawyer.
Your chances are even worse, because you're starting out with some confusion about what a trademark is. A cartoon character isn't a product. A trademark denotes the source of goods or services. Cartoon characters are protected under copyright law and unfair competition law, not usually under trademark law, but sometimes characters are trademarked as well.
You don't state what you intend to do with this character, and that's a fundamental question to what kind of protection it may be entitled to. You need to do some homework about trademarks and character rights, and then see an IP lawyer so you don't waste your time, effort, and money.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
