The ONLY issue is whether the card would hit the drive bays. It's a standard ATX style case so it will fit the other dimensions just fine.
You can open the case and MEASURE yourself because I don't know what case that is.
To be clear, it is IMPOSSIBLE to determine what length of graphics card will fit without knowing the exact model of case or measuring where the card would go length-wise from the rear toward the drive bays.
PSU:
As mentioned you almost CERTAINLY need a new Power Supply. One was listed above for pretty cheap but for a little more I recommend THIS one instead:
http://www.newegg.com/Product/Product.aspx?Item=N82E16817438014
Graphics Card:
I like the 770 but I recommend a different card:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814127741
or
http://www.newegg.com/Product/Product.aspx?Item=N82E16814125463
CPU/APU:
If your computer has that APU in it then just be aware your performance won't be nearly as good as say an i5-4670K for example. An APU has a CPU and a GPU component. When you add a dedicated graphics card like the GTX770 it will use only the CPU part and the addon card for graphics.