I received a DFAS Form 705. It states that during the calendar year 2019 I paid a total of $10,000 toward settlement. How can I claim this on my tax return?

Yes, if you reported this income in prior years, you can claim repayments of over $3,000 on your tax return. You can deduct this kind of repayment under Internal Revenue Code section 1341 as a 'Claim of Right'. 

Deducting a repayment is covered in IRS publication 525 on page 34.  

If the amount you repaid was more than $3,000, you can deduct the repayment as an 'other itemized deduction' on Schedule A (Form 1040), line 16. You can also choose to take a tax credit for the year of repayment. Figure your tax under both methods and compare the results. Use the method (deduction or credit) that results in less tax.

The way to figure these amounts is shown in Publication 525.

To claim the credit you will need to use a TurboTax desktop edition, so you can enter it in 'Forms Mode'.

In the Forms mode, click on Schedule 3 line 12, and it will take you to the  'Other credits and payments smart worksheet'.  Here is where you enter your 'Claim of Right, IRC 1341 credit for repayments of prior year income'.