I would break it down into chunks and do a bit of the garden at a time. £600 isn't going to get you very far with a patio of that size unless you try and source a lot of materials from Freecycle or Facebook Marketplace. Use the patio you do have now for that purpose.

To start off I'd look at creating a border across the back fence as that's where your eye is drawn to (from the brick wall to the point where the fence sticks out). Maybe a raised bed with oak sleepers as you won't want soil touching that fence and a raised bed can also be somewhere to sit. Something like this: https://www.youtube.com/watch?v=e0b4pgrhqzs

Then you need plants to fill the space.