The people at Texas A & M, according to an article in a recent issue of Organic Gardening magazine, have found that the "weed barrier fabrics" can keep fertilizers form getting into the soil, so that is another reason to not use them.
Which cover crop to plant depends on the time of year and waht is available loacally, but Buckwheat would be good now followed this fall with Field or Winter Rye to grow over the winter. Those leaves from deciduous trees would also be a good addition to that soil, as would any organic matter. Clover, Timothy, Alfalfa, Cowpeas, Vetch are some others that could be considered.

i can testify against the fabric...i have heavy clay soil too....all the areas where i layed leaves down and covered with mulch and even applied compost, the soil is loosening up quite nicely and is start to appear much better.

however, around the front of my house, last year, i laid landscape fabric down and covered with mulch, everytime i plant a new plant i have to dig a hole and see that the original soil underneath the clay is still just as compost and ugly as ever....when i decide to redo it, i'm going to remove all the fabric and lay down newspaper, it will suppress weeds during its time, by the time the newspaper rots, weed emergence shouldn't be a big problem.....

outcompete creeping charlie---good luck! I've found CC to outcompete the nasty clover weed, but as a wise one once said, one man's trash is another's treasure.

Here is what worked for me. Start with leaf mold on top. Compost is nice if you have it. Some manure if you have it. Now start chopping and mixing and go as deep as you can. Mix in some more if you can.
Now that will turn to mush when it rains and all the carbon you put in there will suck out all of the nitrogen for the next year, so plant something that does not need nitrogen.
Clover and vetch are ok. If the soil looks yummy to you, consider beans. I would do beans because I like to eat my garden. But I have used each of the other too. Clover was very satisfying and easy.
Then I would do garlic in fall and potatoes next year, all the while mulching and mixing and hilling.

That will be GREAT soil by midsummer next year. Enjoy.