Major in your language of choice at a university. A university education is the most esteemed way of learning a dead language. Many universities offer long-term programs that cover many ancient tongues, coupled with specialty shorter length courses. The University of Dallas gives students the opportunity for a three-month trip to Rome in the spirit of studying Latin. Yale and Cornell both offer Arabic programs.
Study texts written in the language you are trying to learn. Learning on your own can be frustrating, but if you are disciplined, it will pay off. Thomas Flemming of the Fockford Institute says to plan on devoting between five and seven hours per week to study. Don't jump into learning a simpler, more modern version of a dead language. According to Flemming, this is counterproductive to studying the true classic language. Stick to the oldest texts written in the purest form. If the language is well-documented, you will be able to purchase a dictionary that will explain some basic pronunciations.
Use more modern languages to distinguish roots in the dead language. For example, Latin may be considered a dead language, but it is the root language of English and many other languages. Use the roots of more modern words to "sound out" the ancient words.
Find experts in your area and ask them to help you study. This is useful for languages that aren't very well documented or offered at universities, such as many Native American languages. There are still some, though few and far between, who speak or work to preserve these ancient tongues. If it is a Native American language you seek, they may not be living too far away.
Take a quick course, or read an instructional book in the subject to get you started. There are many books you can read to give you a sense of direction. When it seems overwhelming, many people just need a bit of guidance to continue in their learning venture. Websites, such as the Ancient Egyptian Language Discussion List, offer a great deal of assistance also. They have text archives, lists of symbols, discussion boards and a number of other tools to aid the student.