Building a garden on top of asphalt is not as difficult as you might imagine. As long as you have enough sunlight and the spot is not too low, so water won't puddle under the garden, you should have some success. You might find that you actually prefer it to a ground level garden since the soil line is higher and you have to bend over less. A raised bed can also eliminate the risk of contaminated soil leaching chemicals into the garden.
Locate a site where the garden will get from five to eight hours of sunlight per day. Look overhead and make sure that there are not trees overhead that are now devoid of leaves but will leaf out and shade your garden during the summer. Take the time to look at your yard after a heavy rainstorm to make sure that water does not pool where you plan to set your garden.
Place the structure that will hold your garden together in place in the spot you have designated. You can purchase kits from home improvement centers or build your own from wood suitable for outside conditions. There is no need to anchor it to the ground, as the weight of the soil will hold everything in place. Make sure it sits level so that you don't have water runoff problems during the rain. Use sturdy shims or bricks that are strong enough to hold the weight of the raised bed.
Measure your support structure is to determine the size of your garden. Cut out sections of weed cloth to place on top of the asphalt. This will keep your soil from draining out of your garden when you have a heavy rain. Make it 6 inches larger on all four sides so that it becomes a partial liner. Do not use plastic or any material that does not allow easy water drainage or the roots of your plants might rot.
Fill up your raised garden structure. Start with the weed cloth, centering it and allowing the sides to come up on the inside of the structure. Shovel in about two or three inches of pea gravel to keep the soil above the ground. Fill the rest of the bed with good quality garden soil to about 18 inches deep. This will be deep enough for larger plants like tomatoes and peppers. Of course, you can make it shallower if you are just planting flowers or smaller plants like lettuce and herbs.
Water the bed so that the water starts draining out of the bottom. This will settle the soil and get rid of any air pockets that might have developed. You can add more soil if the level drops too low and re-water until the soil level is just an inch or two below the edges of the structure. Allow the raised bed to drain and settle for about 12 hours before planting your seeds and plants.