Dead Silent gaming-PC

I've spent a considerable amount of time researching this subject, not necessarily for your price range but I might be able to provide you with some pointers:

You have 3 realistic cooling options:

1. Silent air (won't be THAT silent, reduced rotational speed = less noise = less performance)
2. Watercooling solutions (expensive)
3. Submerged systems (mineral oil, etc)

1. Finding a decent set of silent air fans is only part of the story. Your specific case layout and chassis will direct air towards points in the case where vibrations could be amplified, or other spots where they could introduce sympathetic vibrations as a side effect. Using fans will involve a lot of secondary vibration damping such as silicone membranes, and resistors to reduce the fan voltage and therefore rotation speed.

Given how dedicated you are to the silence, I doubt this will be silent enough for you. This is without even considering the silence of the PSU...

2. Watercooling is the most realistic and prevalent option out there for hitting that sweet spot of performance and silence. Your two biggest sources of noise with this scenario are the reservoir (which you can eliminate if you instead use a T-junction, recommended for silent loops but very, very messy) and the pump used to circulate your coolant.

Luckily, its quite easy to dampen the pump and isolate a good deal of its vibration from being heard. Simple solutions include mounting the pump inside the case on a block of medium density foam. More complex solutions involve suspending the pump from a hard drive tray using elastic.

3. The most radical option is to buy a case designed to accommodate all your equipment submerged under a non-electrically conductive liquid. Mineral oil is a great example, and you will still have spinning fans but the rotation speed will be negligible. The danger here is dissipating enough heat: most systems involve the upper surface completely exposed (prone to dust) and emitting heat into the atmosphere. More advanced systems include secondary conductive radiators that drain heat away from the submerging vessel (think like heat pipes) to improve the thermal performance of the solution.

My personal recommendation would be watercooling, but depending on how much money you wish to put into such a venture and exactly how silent you want it to be, you could be spending over $1500 on an entirely silent setup. My advice to you is to use what I've said, and what other people will say, as a starting point for your own research. In the end, I can advise you all day as to what I'd do - but you'll have to be happy sinking your money into a solution to get what you want.

Good luck!