Have you determined that the CPU really is the bottleneck, i.e., is it at least 80% busy while streaming or rendering? If not, then the bottleneck (the component that prevents the task from completing faster) may occur because you are using a hard disk instead of a large enough SSD, all memory is used and adding another 8GB would help, etc. What rendering format are you using, e.g., 720p @ 30 FPS?
To be honest, you need to increase your budget significantly to double the performance of your current CPU. Read reviews like
http://techreport.com/review/23750/amd-fx-8350-processor-reviewed/12 (read the whole review if you have time) and then decide on which processor or GPU (depending on the software) best meets your expectations. Please note that most rendering benchmarks use a SSD because a hard disk could become a bottleneck.