What are the research techniques for an American Indian ancestor, great-grandparent?
Research for American Indian ancestors living in the 20th Century are essentially the same as those of any other people in the United States. But, of course, as you go back in time, you may encounter some considerable differences in the number of written records referring to the Indian population. More than any other community, the researcher should learn all that is possible about the history and culture of the specific tribe or tribes involved to locate the various records that may have been created in the jurisdictions for those tribes.
The first step is ascertaining the tribal association. This is accomplished by determining the tribes associated with the locality or localities where the ancestor lived. Quoting from the FamilySearch Research Wiki:
Many records were created by the Bureau of Indian Affairs and their various offices. Many of those records have been preserved by the National Archives of the United States and its regional archives. Some of those records have been microfilmed and are available at the Family History Library in Salt Lake City. Some of the records are also being digitized and indexed by internet websites and by commercial companies.
For a good introduction to this topic, see American Indian Genealogy and the linked pages in the FamilySearch Research Wiki.
Jumping back in time to do the research and focussing on the Indian ancestor before doing extensive research on the more recent generations should be avoided. It is important to focus on the children or grandchildren of the ancestor to make sure that the connection to the Indian ancestor is well established before venturing into records about tribes. Unfortunately for the average genealogist, many of the records of the Indian tribes are confidential and closely held by the tribes. It may take some work to gain access to the records. But there are many more records kept by United States government that are still available.