Components for Building Computers From Scratch/Gaming PC Configuration
I want to build a gaming computer for a budget $1,200. Could I build a PC for this budget that would be better than a regular gaming console like the PS4? If so What hardware components do you recommend? If not What budget do you think I need to accomplish this?
Ultimately this is a somewhat tricky question - while you can certainly build a competent gaming computer for $1200, comparing a gaming computer to a console (especially an unreleased one) is somewhat problematic. The biggest issue you will face is compatibility going forwards - a console is assured to play any software released for it, and the life-cycle of a console is many years (consider the PlayStation 3 as an example); while a gaming PC is not as guranteed (for lack of a better word).
For example, considering the PS3 from above - it was released in 2006, and even software released this year (seven years after its original release) is still compatible, and very playable (for example Grand Theft Auto 5). However even a top-of-the-line gaming computer from 2006 (which would cost more than $1200) will likely not have fared as well - while games from 2006, and probably 2007 (and maybe 2008) will play with no issues, brand new titles will probably either not play, or play with very poor performance. This isn't to say that the console is overall superior - the quality with which a same-era gaming computer will play same-era games (for example The Elder Scrolls IV: Oblivion) is generally higher. You could expect the 2006-era machine to run those games at higher resolutions, with more image-quality enhancements, and so on. Also remember all of the other various things that a PC can do.
Overall it's more of a trade-off - there is no "future proof" with a PC, while a console is "future proof" inasmuch as it will support branded software until it is discontinued. In the case of the PlayStation 4, it's both hard to say definitively what would or wouldn't absolutely better it, and how long it will stay on the market (although history suggests at least five years, if not longer). The longevity makes the console more attractive if you're intending to purchase once, and not deal with any sort of upgrades or replacement (ignoring any potential hardware failure or loss). However if you need other features offered by a PC, a gaming PC can be a reasonable choice as well. Overall if you're on a finite budget, my suggestion would be to purchase a console that offers the games you'd like to play today (and will likely offer games you'd like to play tomorrow as well), and a PC for non-gaming tasks.
Regarding building a gaming PC overall - again I have to stress that there is no "future proof" with a PC - you will generally spend a reasonable amount of money (thousands of dollars) on the initial purchase, and then continue to spend money (hundreds of dollars) on upgrades over time, to ensure compatability with the newest games. It really depends on how much gameplay you get out of the machine, and your personal preferences, as to whether or not this is a good value. For $1200 you can purchase a reasonably powerful machine, but do not expect it to be state-of-the-art a year, two years, or five years from now - just as machines from 2006 are not tip-top today. Spending substantially more does not assure any sort of "future proofing" either - it may give you higher initial performance, and it may mean the machine goes somewhat longer between upgrade cycles, but by itself it doesn't make the machine the "end all" for hardware purchases. Generally I would say that around $2000-$2500 is a healthy top-end for a gaming computer (and any additional money over that should be spent on things like monitors, speakers, etc that can be recycled on future builds, or that contribute to the overall experience; more money into the core hardware usually doesn't do much for you) - that will generally get you top-shelf graphics hardware, a good power supply, and a powerful CPU.
Regarding the "upgrades over time" comment - if you mean to just play one game for the next couple of years (e.g. say you're a huge fan of The Elder Scrolls V: Skyrim, and don't intend to play any other games unless a sequel is released), then it really would not matter if you got a PC or console; whatever they start out being able to run, they will always be able to run until the physical hardware breaks down (which will happen eventually for all components, but in general you can expect many years out of modern electronic devices).
If you have further questions, feel free to ask.