How do I calculate my total tax liability on $750 of nonemployee income? Please breakdown the calculation.

You really have not given us enough information.  Don't know what tax bracket you are in-what other income you have.  

Self-employment tax is $106. add to that your Federal Income Tax.  If that is all the income you have, you will owe $0 Federal Income tax.

It will be added on top of a joint income over 120k minus a standard deduction in 2018.  My wife is having to file a W-9 to get paid for dancing but she will have to divvy it up with 8 other people.