If i was building a computer with your budget for myself this is what I would get:
http://www.hardwareversand.de/pcconfigurator/meinpc/1321322
This is a german site, you can change the language to enhlish in the top right corner, priced in Euros, not sure where your from but I live in Ireland and theyve no problems delivering there. Set price for delivery, around €12 for total order delivery costs.
Specs:
CPU: Intel Core i5-4460
GPU: MSI GeForce GTX 960 Gaming 4GB
MOBO: ASRock H97 Pro4
RAM: 8GB-Kit Corsair Vengeance Low Profile
CASE: Corsair Carbide Series 200R
PSU: Corsair VS Serie V650, Non-Modular, 80+
HDD: WD Blue 1TB 6Gb's
SSD: Crucial SSD 120GB BX100
DVD DRIVE: Samsung SH-224DB
WIFI ADAPTER: ASUS PCE-N15
Monitor: Samsung Monitor S24D300H
They also build the computer for you, assembly fee's sometimes goes on sale, usually €30.00 but drops to €20, €10 and even €0 alot.
Price of PC Specs, Assembly and Monitor without delivery: €1,012
I havnt really looked at the new AMD cards so im not sure if your better off going with AMD, if theres a particular card in mind just do a quick search like "MSI GTX 960 4GB vs 'your choice of card' benchmarks" and see how it stacks up, just keep in mind nvidia usually has better launch drivers for new games, took AMD two weeks after the launch of GTA V to get decent drivers while nvidia had them at launch.
If your mainly focused on gaming, you could probably get an AMD CPU so you can then maybe get a better GPU, but if you do alot of editing/redending/CPU heavy usage then I would stick with the intel CPU, but if you do decide to change, make sure the CPU is atleast 4 cores as some games are start to require it and wont start unless you have them.
The total price with delivery + assemble would be around €1024, if youve no problems building systems yourself then scrap €30 off that as you can turn off the assembly fee and build it yourself.