Disclaimer: I haven't actually built a PC before. However I'm working on a build of my own at the moment (only PCPartPicker builds for now, not IRL yet) and have done an extensive, one might even say obsessive, amount of research (~30 hours on PCPartPicker, /r/buildapc lurking, linustechtips, logical increments, toms hardware, and many many tech forums). Aside from the CPU and GPU, this is almost identical to what I'm planning to build, so it gets the official HELLAchopter stamp of approval. I aimed for well reviewed, high quality parts that might cost a little extra but should give you some piece of mind.
Anyway, here's what we've got:
PCPartPicker part list / Price breakdown by merchant
The cheapest i3 available. It's a dual core, but should do fine in games that aren't heavily threaded (very few games are). If you want to go with an AMD, the FX-6200 has similar performance and can be had for a similar price. The AMD will use slightly more power and some people thing it needs aftermarket CPU cooler. I don't want get into the debate between AMD and Intel, I'll just say that these two cost the same. People smarter than me have answered most of the questions you may have about this choice, I recommend Intel.
A well reviewed high quality board. Some Australian guy on youtube (Tech YES City) loves this thing for budget gaming.
8gb is plenty. G.Skill is high quality and this is a great deal @ $60.
People like Caviar Blue HDs. It's 7200RPM and 500 gigs.
As good as you're gonna do on your budget. Should be plenty fast for what you want to do (gaming not on max settings). Sapphire's are said to be very high quality.
A basic design with plenty of space, 2 fans pre installed, and 5/5 on newegg with 3200+ reviews. Also currently on sale.
80+ Bronze Rated and plenty of power for your non-overclocking needs. There are a ton of options here and they go on sale all the time. If you're vigilant enough you can catch an awesome deal. Make sure you're in the 500w or higher range and 80+ bronze or better.
Plays and burns CDs and DVDs and costs $15.
If you plan on retiring your old computer you can use your current windows license and avoid paying $90 for windows. Your other options are somewhat more pirate-y if u kno what im sayin.
Given your budget I'm assuming you're going to use that TV here. 1080p/60MHz monitors will run you ~100$ and 720p ones ~$75. Check the refresh rate on the TV. If it's 60MHz you don't really stand to benefit from getting a monitor anyway
It's $10, I use one at work and love it. It's also got a ton of good reviews on Amazon. I'll be honest: I don't know dick about gaming keyboards so I'm sticking with what I know.
Sooooooo there we go. Any questions?