Acquire the necessary academic credentials if you do not already have them. A minimum of a bachelor of science degree in Linguistics or a bachelor of arts in English is required. A master's degree or at least most of the coursework toward a graduate degree would give you a better chance of getting hired.
Gain experience as an ESL instructor at your local community college or by working with individual clients. UCLA's ESL program is specialized and will only accept highly qualified and experienced faculty, so you should have a good bit of experience before you apply.
Apply for a position as a faculty member at UCLA to teach ESL after you have sufficient experience and academic credentials. It is likely to be an extensive application process, and depending on the position you are applying for you could be expected to interview with several people.