Translation:I do not want to testify against you.
Not much, and it's hard to translate: nicht modifies the verb or adjective that it precedes, so "Ich will nicht gegen dich aussagen" is approximately "Against you I won't testify," whereas "Ich will gegen dich nicht aussagen" is "I won't testify against you."
It doesn't translate well because in English, "won't" can really only modify the verb, testify.
The following says it goes in front of the prefix, but I suspect it should add "when modifying the separable verb." http://german.about.com/od/grammar/a/The-Position-Of-Nicht.htm
Here's everything you wanted to know, and then some, about the position of nicht (except nothing about separable verbs): http://www.dartmouth.edu/~german/Grammatik/WordOrder/WordOrder.html
"speak out against" is perfectly good English, and matches the German construction. It would be used in more of a public context as a opposed to a courtroom. "I speak out against injustice/corruption/nepotism" as opposed to "I testify against a defendant in court". IMO, the example could mean either one.