Is there any particular reason why you need exactly FFmpeg? I guess it could be possible to get prebuilt binaries, but since it is a C library, you would also need a JNI wrapper code - I don't think anyone would generate that for the whole library since it's quite large, so even with a prebuilt library you still need to have some JNI knowledge to wire through the communication between Java and FFmpeg in C. Just for the purpose of demonstration - here is an example.
Apart from that if there is not a special reason to use FFmpeg, why can't you stick with MediaCodec that is a part of the Android API? It wraps the native StageFright library and could provide hardware support on devices where it is available, while FFmpeg would be a CPU only solution. Of course, if some unpopular codecs or muxers that are not available in MediaCodec are needed, then FFmpeg is the way.