The first thing to look at would be the PSU capability, you failed to mention what is on board, then of course a video card sufficient to the task. 8gb of ram may be sufficient dependent upon what you are going to be doing. The i-5 is a decent processor but the i-7 might have been a better choice if you plan on getting radical.
I'm eventually going to have a dual screen setup. Possibly both being 24" monitors. I'll be streaming a lot of videos games and editing a lot of videos. My question is, what should I change in my current build to comfortably do all the things I listed.
My computer spec -->
Operating System: Windows 7 Home Premium 64-bit
Processor: Intel(R) Core(TM) i5 CPU 750 @ 2.67GHz (4 CPUS), ~1.7GHz
Memory: 8192MB RAM
Graphics Card: ATI Radeon HD 5700 Series