The one thing you have to remember when streaming games over your network, is that network performance is incredibly important. That means if you're going to use wifi and you want hi-def graphics, the actual gaming PC must be connected via ethernet, at a minimum. Then whatever device you're using as the monitor should have a strong wifi connection to the network using wireless N.
In terms of the computer, there are a variety of reasons you could run into a blue screen. Some important things to consider when you're building a performance computer, outside of tech specs of the parts themselves:
1. Cooling (VERY IMPORTANT)
2. Power
When it comes to cooling, this means getting large enough fans to handle your components, and making sure the parts are installed and wires routed in such a way as to allow as much uninhibited airflow as possible. You may consider also replacing the OEM fan on the CPU with a larger one with a good copper core to help as well. The CPU and GPU are gonna be your main sources of big amounts of heat.
As for power, the biggest thing is making sure you have enough. Get a power supply of reputable brand, with not just enough to cover the power of the components you're putting in, but some overhead as well. Shaky power supplies, or ones with not enough juice, can result in an unstable computer, especially during gaming when power consumption is at its peak.
From there, the parts picker will help you figure out what else you need. If you're curious about the general effect of different parts on the gaming performance, here's a rundown:
GPU -- your main workhorse. Unleashes the ability to unlock all those amazing graphics. Easily the most expensive component. Read reviews to see what works for you and your budget.
CPU -- a close secondary workhorse. If you want to be able to max out everything, you'll need a high end CPU as well, because for as much as GPU's do, most of the information when running a game still needs to pass in and out of the CPU. So if it can't keep up, you'll get stuttering graphics and won't get that nice, smooth 60 fps everybody wants. For people trying to make a budget gaming PC (like myself), you can save a little bit of money on a CPU, but go too far and you'll just cripple your nice shiny GPU. Also, from everything I've read, if you're really serious about gaming performance, you have to go with Intel. AMD just doesn't keep up on a per-core basis.
RAM -- You just have to have enough. More than enough doesn't make a difference, but with too little it'll hurt performance. 8GB at least, 16GB if you can afford it.
Hard Drive -- A faster hard drive will mean faster load times between levels, or when opening the game. For some games that don't typically have a lot of loading screens, this may not make much of a difference. And certainly during normal gameplay it doesn't matter all that much. In this space, a 7200 RPM hard drive will easily give you the best bang for your buck in terms of a decently fast hard drive with plenty of space, since games take up a lot. Purchasing a Solid-state hard drive would be considered a luxury, only if money is no object, since getting a large SSD is still very expensive (although cheaper than they used to be). The SSD would speed up loading times to the point of nearly wiping them out, but again, it doesn't change your graphics or gameplay at all, so definitely a luxury.