It is very difficult to prove discrimination, as there can be many legitimate reasons not related to race for the company's failure to promote you. You should, however, speak to an experienced employment attorney as soon as possible--many of us will provide a free consultation.
This answer is provided for informational purposes only and should not be relied on as legal advice. You should be aware that no attorney-client relationship is established through this answer and none will be established without a personal consultation and the signing of an engagement agreement.
Frankly, you don't seem to have enough facts to do more than speculate as to the reasons for the fact that they didn't offer you the promotion. Proving a case requires showing either an overt act or a pattern and practice. An example of an overt act would be telling you that you're just not good enough because of your race. A pattern and practice would be several other people who were refused promotions and who were of a different ethnic background than your boss. It's often harder to prove cases for professionals since there are fewer of them in most companies.
Have you considered meeting with your boss and asking him in a completely non-confrontational way why he seems to be looking for someone other than you? You may find out that he feels you're lacking in some important criterion and that can give you a direction on where to focus improvement efforts. And, it may give you a leg up on the next promotion opportunity. Just something to think about.
Hope this helps.
Please be sure to mark the best answer to your question. My answers are general and do not form an attorney-client relationship. I'm happy to talk to prospective clients in my areas of concentration and geographical location.