Point 1 won't be any trouble at all. Any decent GPU will have HDMI output on it. Point 2 is a lot harder on your budget. You'll need a high end i5 or low end i7 to run GTAV smoothly at high FPS, for refence; I run a i7-4770 which is mid range for the i7 and it costs roughly $350, a decent i5 costs $250. Expect to pay another $150 for a decent GPU and about $75 for 8GB of RAM, that's $475 already.

 

Thankfully, there's are plenty of places to save a bit of money; if you can borrow a DVD rive of a buddy you can use that to install the OS and avoid having to pay $50. With platforms like Steam you won't need a DVD drive to install games anyway, I can't actually remember the last time I used an optical drive to install anything other than the OS. A tower can also be bought on the cheap, as can most internal wiring. A simple 500GB HDD can be a cheap as $50 and you can always get more HDD's later on.

 

Don't skimp on the PSU and mobo though. A cheap PSU can fry all your expensive part and kill them dead if it fails and buying a cheap mobo can be done but you'll be buying a new one next time you want bolt any new parts on.

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