Just to be clear, the crackling sound is coming from the computer components, as in the PSU, the motherboard, the video card, or a fan - right? Those sounds are not coming out of your speakers, correct?
If that's the case, it doesn't sound like an audio problem with the sound output. Start by disconnecting the speakers though. Pull the audio connection out (with the system off). Then, one at a time, pull the power from one fan (system off), boot up and listen for the problem, and then proceed to the next fan. Include the CPU fan but plug another fan into that CPU fan header and set that fan outside the case. The heatsink is adequate to keep the CPU cool enough for a few minute test. You are looking for a mechanical source of the noise, like a bearing. If you find that one fan is causing the problem, that's a cheap solution to fix it.
Now, if it's the video card causing your issues, you would need to swap it out with another video card (maybe from a friends computer) to see if the noise goes away. If that's not the cause, then it could be an arcing of one of the cables to the case, or even the motherboard arcing underneath to the case. If I got to this point, I would pull all the components out of the case and hook them up on the table (use the motherboard box to support it and the video card). Inspect all cables carefully and look for burn marks. Inspect the back of the motherboard carefully too.
If you still hear the noise with the system running outside the case, try your best to keep the parts separated so you can determine which part is making the noise. At this point I would really focus on the PSU, but it could be that video card.
On a side note, your video is good in that I hear some crackling noise, but I can't tell where it's coming from as you move your phone around. Be careful with those fans and getting your fingers in there.
As Albert Einstein said, "I could have done so much more with a Big Al's Computer!".
