curious about the  cpu? any good? Hoping to play games like skyrim, saints row 3, MMO's, arma 2 or 3. Priced under 600. Would upgrade video card for sure. Need a computer asap for work. Cant afford much higher.

The CPU won't set the world in flames, but for now it should do the job well enough. The amount of RAM is a bit weird. Besides that, it doesn't look bad.

Are those 600 pounds? If so, it's a bit expensive for what it is. If you can/want to build your own PC, this would be a good guide and for less money

http://www.bit-tech.net/hardware/buyers-guide/2013/08/22/pc-hardware-buyer-s-guide-august-2013

But it doesn't include OS.

600 dollers, the processor is always upgradable right? Like as in adding a amd fx 8 cote. Adding processor and graphics card it should make for a good gaming pc?

Make sure you have big enough casing (for both air flow and space) and motherboard if you're planning on upgrading. Tbh if you have the knowledge why not build one yourself? You can get better components at that price point if you assemble one yourself.

Anyway check out this thread http://www.neogaf.com/forum/showthread.php?t=509570

Edit: I would go for intel processor over AMD, especially core i7 and i5 overclock like beasts. 

You can get something better than that for the money. You can check my thread http://gamrconnect.vgchartz.com/thread.php?id=168659&page=1
There are some builds on you budget on that OP

No, you would not be able to upgrade that to an 8core. The 8core would have a socket AM3+ motherboard, whereas the A10-5700 has a socket FM2 motherboard.

In other words, to upgrade that processor to an 8 core, you would wind up purchasing a ~$60 motherboard as well as the processor itself.

If you want a quick glance over performance top down for graphic cards, reference this chart.

http://www.tomshardware.com/reviews/gaming-graphics-card-review,3107-7.html

http://www.scan.co.uk/
http://www.novatech.co.uk/

These guys have amazing prices for components and next day delivery. Both have top notch customer service.

What if I leave the processor and upgrade the graphics to a Radeon 7870 hd. Will she be able to run skyrim nicely and smoothly?

The problem with your CPU is that it will bottleneck that GPU(7870 and above). You should get an AMD 6300 or the cheapest i5 you can get. Then grab a 650 ti boost/hd 7850 or a hd 7950 for a little bit more. You should be set for the next years no problem