First you will want to prepare your resume. Make sure to include special skills, education, training or job experience you have that would make you a great trader. Be aware that although a degree is not officially required to trade on Wall Street a large majority of people working on Wall Street have MBA's or law degrees. It is possible but very rare to obtain a trading job without a degree.
Once your resume is prepared you will want to begin searching for an opening as a Wall Street trader. There are not an endless amount of job positions as traders and you may have to wait a while until one opens up. At first you may have to be satisfied with any job on Wall Street that you can get, even as a runner. It may seem frustrating to have an MBA and to have to accept a position as a runner, or a number of other entry level positions, but just consider it a foot in the door and keep your eye on the prize. Try talking about your career goals to your friends and family. They may know of a trader job that you don't. Many people find work through someone that they know.
When a spot does become available and you are given an interview make sure you show up on time and dress nicely, preferably in a suit. Wall Street is a very image conscious environment and you will want to dress the part. Also make sure you know what is currently happening on the different exchanges so you can speak intelligently about it at your interview.
Good luck on your search for a Wall Street trading job.