Building a computer from parts is no harder than putting together a piece of shelving or any other easily done home project, it's really just a matter of putting the right pieces in the right places. It really is that simple. Once the parts are in the right places and you're able to flip it on, inserting a windows disk and running the setup will pretty much take care of the rest. Once upon a time you'd have had to worry about BIOS and such, but nowadays that's all taken care of automatically unless you really want to tinker (Which I would suggest against unless you know what you're doing!).
I'm using a computer that's bordering 5 years old now and it's still capable of running just about anything out today on near maximum settings. This long console generation has been good for me in that since so many games are built with the 360 and PS3 in mind, even a relatively archaic computer like mine is considerably more powerful than either console.
You could easily build a computer much better than mine for under 1k, including the monitor and accessories. Still more expensive than your average console, but when you also take into account the amount of money saved on buying games thanks to services like STEAM offering such good deals and mods/clan communities adding life to older games which would since have been abandoned on consoles it quickly evens out. Games in general even retail are generally cheaper than their console brethren as well, which is nice.
Still, I wouldn't recommend PC gaming for everyone. It's a bit of an investment, and quite often running older games and hell, even newer games sometimes can require a bit of running about and troubleshooting. Requires patience. And if really you're just looking for a 'pop the disk in and play' experience then the PC probably isn't the platform for you. Nothing wrong with that, simply a matter of tastes. It's also why I'm so disgruntled that process such as patching and even installing have become common on consoles. I play console games to get away from that stuff.
But anyway, there you go. If you want specific hardware recommendations I'm sure there are plenty much more knowledgeable than me who could help.
Good luck and whatever you decide, have fun!