This Franklin lineage tests into the FY02 group, haplogroup R-CTS6919 of English ancestry and therefore is definitely not Native American. As the two noted claims were both rejected and there is mention of a Green connection, it is highly suspect now that these Franklins will somehow also be connected to John Franklin and wife Sarah Green, a reported descendant of Gardner Green who had some 300 descendants file claims of Native American ancestry coming from him (see Sarah's bio).
Henry was born in 1798 in North Carolina. He was the son of Henry Franklin and Elizabeth "Betty" Gibbs.
In 1820, he lived in Buncombe, North Carolina. 
↑ Franklin, Ben. Franklin Family Researchers United. Volume 16, p4-16. Note: While this forward lineage may be accurate the claim of a Franklin full blood Cherokee ancestry is not, and is likely misattributed from the Green lineage as noted in the bio.
↑ A source is needed for this date and place of birth.
↑ From the application #21178 for Henry J Franklin (see also application 44698 below), NARA M1104. Eastern Cherokee Applications of the U.S. Court of Claims, 1906-1909. Roll: 0183, Application 21178: Henry J Franklin. Descent from parents John Franklin and Mary Mapels, grandparents Henry Franklin and Tabitha J Gibbs, great-grandparents Henry Franklin and Betty Gibbs.
↑ "United States Census, 1820," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:XHG9-YVP : accessed 13 April 2018), Henry Franklin, Buncombe, North Carolina, United States; citing p. 70, NARA microfilm publication M33, (Washington D.C.: National Archives and Records Administration, n.d.), roll 80; FHL microfilm 162,796.
Franklin, Ben. Franklin Family Researchers United. Volume 12, p4-16. Article entitled Cherokee By Blood, wherein appears extracts of Franklin applications to the US Court of Claims 1906-1910 from the work Cherokee By Blood, Records of Eastern Cherokee Ancestry compiled by Jerry Wright Jordan (1987). Application #21178 of Henry J. Franklin b.1845, living at Madisonville, TN at the time of application. Son of John Franklin, claiming 1/4 Cherokee Indian who married Mary Maples and grandson of Henry Franklin, claiming full blood Cherokee who married Betty Gibbs. Application REJECTED.
Eastern Cherokee Applications of the U.S. Court of Claims, 1906-1909. NARA M1104. Roll 0340. Application Number 44698: John Franklin. Descent from parents Henry Franklin and Tabitha Gibbs, grandparents Henry Franklin and Betty Gibbs, great-grandparents John Franklin and Betty Green. Rejected. (if someone can upload a copy, please do. Thanks!)
It may be possible to confirm family relationships with Henry by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
Y-chromosome DNA test-takers in his direct paternal line on WikiTree: