I've paid a relative to watch my child but they do not own a business. Can I still claim the child care credit?

Yes. You can claim this on your taxes as long as they are not one of the following type of relatives below. You will need their social security number or they will need to get an EIN number. 

They can not be:

• Your spouse;
• Your dependent;
• Your child if they were under 19 on the last day of 2015, even if not your dependent; or
• The parent of the Qualifying Person, if the Qualifying Person is your child under the age of 13 during 2015.

https://ttlc.intuit.com/replies/3301418

How to get an EIN number:

https://turbotax.intuit.com/tax-tools/tax-tips/Small-Business-Taxes/How-Do-You-Get-a-Business-Tax-ID...

And if you issue a 1099 to the relative, they will have to report the income

As long as you have proper documentation of the expense and the childcare was for you to get to work, it is deductible. It does not matter that your relative is a "dba" or unincorporated.

If the relative is the other parent of the child, you may not be able to take the deduction, but aunts, uncles, your brother (or in-law) sister, grandmother/father are ok.