If the termination is granted, all of his rights will be terminated including the obligation to pay support. You need to have lived in Texas the past six months to file. A Termination is completed and should be done with the assistance of an attorney
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His obligation to pay support may continue if the Attorney General is involved in the case or if the child is receiving/has received state benefits.
There is not a way to file voluntary termination papers on your own. I recommend consulting a family law attorney to help you prepare the papers.
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A suit to terminate the parent child relationship will require the assistance of a private attorney unfortunately and will cost some money. Once the termination is granted, he will have zero rights to your daughter and in all likelihood will not be obligated to pay anymore child support.