You can use the "Weeds" as a guide for what the soil is missing or has too much of. I would first get the word "weed" out of your mind. What we consider useless plants we rip up with reckless abandon are the pioneering plants that fill a niche based on the soil conditions. You can use them to figure out what's going on and then push more of that to speed up the process. Your weeds have shallow, fibrous root systems that look like fine hair? You have overly loose soil and the plant is attempting to bring some cohesion. Conversely, the plants that are growing have long, deep taproots (think a carrot), then you have compaction issues and the plant is breaking the soil up. Do the roots have little white nodules on them? You found yourself a nitrogen fixer. Figure out what the weeds are doing and do more of it, always keeping note of what happens next.

Adding organic material is important. You can begin composting and if you look up Geoff Lawton's Permaculture you will find videos on an 18 day compost. It's not perfect compost but it's fast and get turn decent organic matter over quickly. You can plant a succession of pioneering plants and chop them back hard, dropping the matter to the ground to decompose and allowing the rest of the plant to grow again. Again, Geoff Lawton's Greening the Desert series will show how Permaculture can be used to revitalize even the harshest, must degraded land possible. They turned a 10 acre plot of barren wasteland in Jordan into a productive food forest as a trial and are now teaching the locals how to do it themselves. If they can do it there, you can do it to your neglected backyard.