Well, for most mh/s v.s. $ the 5830 wins hands down. ( 280-300mh/s)
the real question is to get how many.
the general rule of thumb is that 2gb of RAM, a Sampron 140 CPU is standard business due to the seer price/clock speed ratio it has.
since this is NOT a dedicated mining machine you are welcome to bump it up the ram a bit, but the amd S140 OCs very well and should not be a problem for the light usage you listed.
The real question when comes down to building a Rig is 2x 5830s or 3+ 5830s.
each 5830 uses about 180W of power under full load, but as PSU price doesn't scale as well as slots, the choice is up to you.
for the 2x 5830 set up (one of my mining RIGS)
a 600W PSU of your choice I personally have http://www.newegg.com/Product/Product.aspx?Item=N82E16817139028
due to the fact I just don't trust those lesser known names (there's a $20 one with over 300 reviews that's pretty good on newegg, again; this is just me)
2x 5830 http://www.newegg.com/Product/Product.aspx?Item=N82E16814102878&cm_re=5830-_-14-102-878-_-Product
Totaling ~450-500 bucks(shipping included etc), estimated mh/s 560(low), 600(high).
For the 3-4 5830 set up depends how many 5830 you end up putting in your board.
for 4x I recommand AT LEAST 1kw PSU.
4x 5830 will set you back around ~$800 which is your budget
this of course doesn't include things like monitor, keyboard/mouse, dvd drive to install OS, or HDD.
I highly recommand Linux due to the seer Mh/s Hashkill(a linux only miner) produces. but I won't go into detail unless you ask me bit more specific questions.