$1000 for a Core i7 and a 4870x2 is VERY limiting for other components. The cheapest Core i7 CPU is the 920 for ~$300, and the cheapest 4870x2 is ~$400. The cheapest LGA 1366 mobo is ~$200. So... you're left with $1000-$300-$400-$200= $100 left for a case, power supply, and memory. There's no way you'd be able to add a monitor or operating system, so I'm assuming you already have those, or a free OS like Linux or something.
Anyway, here's the best I could find for keeping within your budget limits.
The aforementioned CPU and Vid Card:
Core i7 920- $300
Powercolor Radeon HD 4870x2- $400 after MIR
Gigabyte GA-EX58-UD3R motherboard- $185 after MIR
GMC R-2 Toast AZT-GMCR2-BL Case- $25 after MIR
PQI Power series 1GB DDR3 1066 RAM- $23
Raidmax Hybrid 2 RX-530SS 530W PSU- $40 after MIR
Total- $973+shipping
And you have to deal with Mail-In Rebates (MIR) to get that price.
I'm not sure the vid card will even fit in that case. You really would want a full tower case, but they're ~$60+. Also, I'm sure that PSU isn't certified for that vid card, but it should work. You really would want a decent 650W+ PSU, but again, they would be ~$60+.
And remember, this is without a monitor and OS.
It would be much better to have around a $1250+ budget for a Core i7 setup, especially with the 4870x2; $1500 with a monitor.
You can check out my "My i7 Build" post for a decent Core i7 setup for around $1700.