Well, it's authentic at least in the sense that it did appear in several U.S. newspapers in the last third of 1865. I've uploaded a copy of the letter as it appeared in The New York Daily Tribune (as well as a page view so that you can see the masthead) on August 22 (p. 7), which represents the earliest printing I've found. (I don't have access to The Cincinnati Commercial before 1867.) For comparison, the letter also appeared in The Agitator [Wellsboro, Pennsylvania] on October 25, 1865.
I don't really think this letter is a fabrication, but I do suspect that Anderson's letter was likely a collaborative effort between the former slave and a party seeking to give support to the post-war freedmen's movement (hence its relatively rapid appearance in The Cincinnati Commercial). Whether this help entailed simply transcribing Anderson's reply or whether this person assisted in reworking the letter (possibly with Anderson's knowledge) is unclear. (Again, I'd be interested to see the first letter, presumably sent by Col. P.H. Anderson of Big Spring, Tennessee.)http://i14.photobucket.com/albums/a319/Cackalacky/Page.jpg
As further support of the authenticity of the letter and its contents, I direct the reader to the 1870, 1880, and 1900 federal censii for Dayton, Ohio which show Jordan Anderson (b Dec 1825 in Tennessee) in a household with his wife Amanda (b Oct 1829 in Tennessee). In the 1870 census, five years after the letter was published, they were listed with four of their children -- 19 year-old Jane, 12 year-old Felix (Grundy?), 5 year-old William, and 1 year-old Andrew. Over the years, Amanda had had eleven children, only six of whom were still living in 1900. Three of the children we were living with them 1900, including their 29 year-old son Valentine, a physician. In the years of the censii, Jordan lists himself as hostler, a coachman, and a butler. He cannot read or write, and Amanda can only read, but all of his children attend school in the records shown.
Patrick Henry Anderson Sr., born 1823 in Tennessee, merchant and farmer of Wilson County, Tennessee, appears in the federal censii of 1850 and 1860, with his wife Mary Ann, and his children Patrick Henry Jr., Martha, Pauldin, Timis, Edgar Poe (Allen?), and Mary. The slave schedules of 1860 show him as the owner of thirty-two slaves, including a 34 year-old male who could be Jordan. There's a three-year old boy who could be Felix and a ten year old girl who could be Jane, but Amanda doesn't seem to be in the list, unless her age has been mis-recorded. As genealogists will know, slave schedules did not include the names of the slaves, just their age, sex, and whether they were black or mulatto (of mixed ancestry). Notably, seven of the slaves, all of them minors, were listed as mulatto, however the distribution of ages of slaves (in particular the lack of female slaves of the correct age to be mothers) suggest that many of the younger slaves came from different owners originally.
According to other published and online records of his family tree, P.H. Anderson died in 1867. His son, P.H. Jr, the Henry mentioned in the letter, appears in censii in Wilson County as late as 1880.
There are multiple George Carters in Wilson County in the period in question, but the likely one is a carpenter who appears in censii in 1850, 1860, and 1870 in the same township as the Andersons. Before the war he owned two slaves, and each was mulatto.