You can play LoL with all graphics settings maxed at 1080p with a GTX 650 Ti.
League Of Legends: High Details, 5760x1080 : League Of Legends Performance, Benchmarked
Though if you're picking up more graphics-intensive MMOs (eg. Neverwinter), a HD7850 2GB or higher would be feasible. Depends on what resolution your monitor will run at, really.
Also, an Ivy Bridge i3 will suffice.
edit:
From PC Themes pricelist:
Intel Core i3 3220 + Asrock B75 Pro 3 M - $256
G.Skill Ripjaws-x 1600MHz CL9 4GBx2 - $95
Plextor M5P 128GB SATA-3 SSD - $169 (for programs/OS)
Seagate 2TB 64MB SATA-3 HDD - $117 (for data storage)
24X DVD-RW SATA - $25
Powercolor HD7850 2GB PCS+ (with Never Settle bundled games) - $267
Seasonic M12II 520W - $99
Xigmatek Midgard II ATX Case - $89
On-site assembly - $10
Total: $1127
Never Settle bundle game list:
http://forums.hardwarezone.com.sg/h...eloaded-details-asus-power-color-4117500.html
If you really want to spend the whole budget though, change the i3 to an i5 3470 (should be around $100 extra), and a better card (if your monitor is 1080p or above; otherwise get a better monitor).