I'd start with this month's $500 SBM build. Get an i3-3220 CPU instead, and a 2GB version of the HD7850, possibly one with a quieter cooler if that matters to you. You'll be under your budget (unless you also need Windows), but have a nice gamer. Unless you need wireless internet, the motherboard has a RJ45 Ethernet port on it. If you need wireless, you should be able to pick up an add-in card for ~$25. USB dongles are even cheaper, but they can be slower, which might not be good for games, especially twitch shooters.
Onus has it spot on. There's still the question of what to do with the rest of your budget - you could obviously save it, but there are a few more luxuries you could afford with the remaining ~$70ish:
- you could opt for a solid-state drive (or more mass storage). You won't notice terribly many benchmark score increases, but I'll be damned if your computer won't feel twice as fast with an SSD.
- you could get a nicer power supply. The Antec VP-450 is a great PSU, don't get me wrong - but it's non-modular, and there are a number of more efficient power supplies out there (lower heat, less noise, lower power bill if energy is expensive for you).
- you could choose a nicer case. Cases can easily last through multiple builds, so it's always a good idea to get something with the features, quality, and aesthetics you like.
- you could go for more performance, as well; you might be able to afford the jump to a 7870.