If you wish to look for court records in Tarrant County check out the "Tarrant County Records" link below. Each county in Texas has a similar search function. Simply search for the county clerk for each county. Other than that your family member should pull a credit report, which you can do for free at "annualcreditreport.com".
Looking for court records may not accomplish your goal of finding out what the debt situation is for someone else. There will only be a court record if the creditor has filed suit. By the same token, even if you get permission to look at your family member's credit report, many payday lenders do not report debts to the credit bureaus. This lending system is, for the most part, under the radar of the credit system. As a practical matter, if you have permission to help, or are willing to do whatever it takes to obtain this information, I would suggest searching the person's car or looking for their secret paperwork stash (often in their underwear drawer) for the loan papers. Of course, if you pay for this person's cell bill, the call history records may also reveal these secrets. Hope this perspective helps!
If your concern is that the debts were taken out in your name, you need to check your credit reports immediately. You can look county by county, but not all counties are on-line. If you are concerned that you might be responsible for the debts after your relative's death, don't worry, you are not.
This comment is given for educational purposes only. No attorney-client relationship exists between us.