Will the AMD HD 7790 be compatible?

The parts should all be compatible.
The problem is the included power supply is rubbish.
When this fails it could take other components with it.
You can't get a decent power supply this cheap.

I would recommend getting cheap case without a power supply and then buying a decent budget supply like the Corsair CX430:
http://www.amazon.co.uk/Corsair-Builder-Series-Bronze-Supply/dp/B009RMP14M/ref=sr_1_1?s=computers&ie=UTF8&qid=1390907870&sr=1-1&keywords=corsair+CX+430

This would not allow you to upgrade to any video card much better than the HD 7790, at most the Nvidia GTX 660 or AMD HD 7850.

You could go for the CX500 if you wanted to allow for a better graphics card upgrade later:
http://www.amazon.co.uk/Corsair-Builder-Series-Bronze-Supply/dp/B009RMP2VE/ref=sr_1_1?s=computers&ie=UTF8&qid=1390908258&sr=1-1&keywords=corsair+CX+500

This would allow you to upgrade to a 220W graphics card later if you wanted (e.g. Nvidia GTX 760 or AMD R9 270X)