Well first, I'm just going to say that building a custom computer is arguably like adult Legos, and can lead to more for your money. At the same time it isn't punishing to get something pre-built, but it'll cost a bit more.
Speaking of money, you're going to need to give yourself a price range, and also find to what degree of awesomeness you want (Super overpowered computer that can run two games very well and in high detail at once, something top notch with excellent recording capabilities and game FPS, something reliable with a ton of storage for recordings and many downloads, all those sorts of things). I've used this gaming laptop for years and up until a recent cleanup I had 8-12 FPS in Minecraft, running an i3 Core and Windows 7 with 420 GB on my C: drive. All in all, this is just a 3-year-old $350 Lenovo, but can play Steam single player games fairly well.
If you're recording, again, you may want more data space. I don't know any specifics on what computer to get, but something at or above i4 Core and around $700-1000 (including mouse and keyboard) should be good and proper. Don't forget monitor(s) and all that shaz. If sound from the system is something of concern, then choose a computer with liquid cooling etc.