How do you go about unlicensed tattoo artist lawsuit?

It depends on the reason for the lawsuit. If you knew the tattooist was unlicensed before you had a design done, you can't expect to win a case against them - as you chose to enter their business and have the work done. Sure - the tattooist could be prosecuted for unlicensed practice - but that case would be brought by the state - not an individual customer.