You should look into the "Ruth Stout" method of gardening. Basically she piled on mulch so thick it choked out the weeds and enriched the soil.

I've been working on getting that right. I take all my leaves and grass clippings and pile them on my garden and around the plants. That's helped a lot, but I still had weeds take over by August last year. Ruth Stout said that if you have weeds you need more mulch. I piled more on this year than ever before, but I'll still have weeds. I still need more mulch

She also said, "don't worry about weeds". If you get a few weeds it's not that big of a deal. Some will deter pest, others will attract beneficial insects, and you can still harvest what you've planted. I went with that last year because it was too darn hot to go out and pluck weeds in August. But I did still go out and harvest what I'd planted and I got a lot out of that weedy garden. It wasn't that pretty to look at though

Another thing Ruth Stout said was that if you kept a thick layer of mulch you wouldn't need to till your garden. This is true. After two years of tilling in leaf mulch into the hard clay soil we have here each Spring and then piling on the mulch all year, I did not have to til this year. I pulled the mulch back and I can sink my hands into what is now rich black soil that's full of earthworms. And, she said, because of this the weeds you have will be easier to pull out. This is true too.

I've basically followed her methods with this plot from the beginning. It's a long process to get it all working right. At least it takes a few years here to get the soil built up and really soft and loose, and years of mulching to get the weeds finally smothered out. I'm not there yet, but I am getting closer every year to really following her advice and so far it's working like she said it will.

I've learned more from reading up on Ruth Stout than just about anyone when it comes to organic gardening. She also mentioned that it gets a lot easier once you get it all going, I hope she's right about that too