Ask Mr. Smarty Plants

Cynadon dactylon, bermudagrass. Guess what? We don't like it, either. It is native to Africa (not Bermuda) and has become one of the most invasive weeds in the South. See this University of California  Integrated Pest Management article on how to get rid of it.

We wouldn't recommend over-fertilizing anything. It's chancy whether it will cause the plant to "grow itself to death" which we have heard of, too. And the excess fertilizer will be right there available to wash off into the street and straight to your rivers and water supply. On the subject of excluding the light from the plants by  black plastic, here is a quotation from the article above:

"Mulches of black plastic or geotextile landscape fabric can also be effective over large areas if light is excluded."

Whatever you do, remember it will not be quick and easy. The reason bermudagrass is so invasive is that it can spread above and below ground, via stolons (aboveground) and rhizomes (belowground). You don't just do it with one operation, because still unsprouted seeds are likely still in there, as well as the rhizomes, which can hold nutrients for the grass to rise again long after you thought the top part was dead.