I'd say it depends upon your gaming resolution, and that creates the biggest problem for you.
At resolutions 1920x1080 and above the A8 5600k APU will push some powerful video cards ... but your budget only allows an HD 7790 -- which may struggle with some titles at that resolution.
But, with a strict budget, you have to do what you have to do. Just be sure to get an FM2+ motherboard.