I use a simple hand tamper, in layers of 2"-3". Actually just did this last weekend. You want to be careful near the foundation as that is pressure on the basement walls if you try to compact too aggressively.

 

I dont have a basement so that is not an issue. I actually found a 10" x 10" hand tamper sold by home depot for $40 weighing 13 lbs and decided to try it. I understand what your saying about tamping 2-3" at a time but the dirt is already in place so I am just going to have to tamp it the way it is an hope for the best.

 

If you plan to place a concrete slab over the soil, you are going to want the correct soil in place. Proper soil to support a slab is well graded sand and gravel, with a limited amount of silt allowed, but no organic soil, and no clay. Of course people build over improper soil all the time, but the result can be cracking of the slab, settlement, or heaving if you are in frost country. Since you have gone to so much trouble to prep the site, it would be unfortunate if you tamped poor quality soil, later put a slab on it, and the slab performed poorly.

 

I live in southern calif. How many inches of this kind of soil do I need?

 

Your building inspector should be able to help you out. Since you are in southern California, probably no frost. Typically you would need to remove all the organic soil (too compressible to support a slab), then place around 6 inches of crushed stone, maybe a little less if the natural soil is sandy or gravelly.