What you can and should do is ordinary family history research and documentation. In fact you won’t get anywhere without having documented the basics.

At some point what you MAY find it possible to THEN do is follow whatever male surname is native american to a living male descendant of that surname line and do yDNA testing. That could be your father, brother, father’s brother, your grandfather, or perhaps a cousin so long as it’s all the same male surname descent.

The alternative for females must be stronger documented evidence from female (mother) to female (daughter) mtDNA testing – but then a mother is certain who her child is, to put it nicely.

The best part about it should you do DNA testing is that you don’t go further than your mailbox. It’s a ‘buccal’ test, swabs of cells inside your cheeks.