You have two different contractual obligations and issues going on here.
First, you have a contract with your pool company. It sounds as though you allege that they have breached the contract and are not entitled to be paid. This is a dispute between you and the pool company.
Second, you have a contract with your credit card company. You should immediately dispute the charges and supply whatever documentation you can to your credit card company. My guess is that unless you have some sort of consumer protection plan on your credit card, you are obligated to pay for all charges you make - regardless of whether you regret it later or dispute the quality of work. I think your chances of successfully challenging the charges depend upon whether the pool company acted fraudulently, or whether there is simply a good faith dispute over the finished product.
You should speak with an attorney to see whether locating and filing suit against the pool company makes any sense and help you negotiate a settlement with your credit card company.
I am assuming that the pool company is not collectible because it is out of business. If this is the case, you should immediately dispute the balance with your credit card company. The law that applies here is known as the Fair Credit Billing Act. It allows a credit card holder to dispute a charge based on the failure of the merchant to have not properly or fully provided you with the goods or services you charged. However, the law provides that you must inform the credit card issuer of the disputed charge within 60 days of the date of the charge. If the 60 days have expired but you have only recently learned that the pool work was defective, you should make that clear when you register your complaint/dispute.
In order for the credit card issuer to give some validity to your claim, you will need to document the defective nature of the work. This can only be done effectively by way of the written opinion of an expert. I recommend Trip Knox of Northeast Aquatic Design at 978-767-8711 (neaquatic.com), who could inspect the pool and give you an opinion as to the quality of construction. Photos would also help.