Classics majors at UC interested in teaching high school should look into Second Language Studies program for foreign language teacher training in the College of Education.
In the UG handbook for Classics, see the following for the various ways you can get certification.
(The handbook suggests that students switch colleges to CECH, but this is not necessary. It will be better for students to have a Classics degree than it will be to have an education degree.)
Remember that even if you are going to teach Latin or Greek at the secondary level and do not have plans to pursue a Ph.D., you still want to have a very high proficiency in the languages. In other words, be sure to be well trained and not just adequately trained.
Almost all schools (public and private) now require their teachers to be certified or at least be pursuing certification. It is possible to get a job with no certificate or experience, but having an M.A. helps. If you are able to get a job before you are certified, you may be able to bypass student teaching and only have to take courses/exams for certification.
Join the American Classical League and CAMWS. They both offer help/funds to teachers. The ACL website posts available jobs--by far the best place to look. Check it daily when looking for work. The ACL has lots of good stuff on their site.
Suggested Reading: The Built-in Defects of American Education by Paul A. Zoch (a Latin teacher from Texas); The Art of Teaching by Gilbert Highet.
It helps to be qualified to teach another subject as well: consider a minor in English, History, Math, a modern language etc.
Search the web for questions typically asked during teaching job interviews. Be sure you can say why you want to be a teacher and why you want to teach Latin. Be confident in the interview but not pretentious: remember that you are dealing with real-world administrators, not eccentric, ivory tower academics.
Familiarize yourself with all of the textbooks and approaches. There are two main approaches 1. The Traditional, Grammar-Translation approach and 2. the so-called "Reading Method" approach.