Joseph apparently made several copies of the characters from the plates, possibly by placing a sheet of paper on the plates and rubbing it with charcoal, and then having someone trace the impressions. Both Emma and her brother Reuben assisted, as well as Martin Harris. A copy of the characters published in The Prophet, the New York-based LDS newspaper, in 1844 has some variations from the more commonly depicted version, corroborating the claim that there were multiple copies. (See Chapters 2-3, From Darkness unto Light by Michael Hubbard McKay and Gerrit J. Dirkmaat.)
Martin Harris took a copy of the characters to Luther Bradish, who referred him to Drs. Anthon and Mitchill in New York City. They were both interested in Native American Indian stories. Mitchill had learned Mohawk language and remained interested in the origins of the Indians, relying on Humboldt's theory of three separate sources of Indians. Anthon was collecting Indian stories about the time when Harris visited him, so of course he was interested to see what Harris had to say.
A lot has been written about these events and the characters. Harris came away convinced that Joseph was telling the truth. He mortgaged his farm to get the Book of Mormon published.
By contrast, Anthon claimed years later that he told Harris he was being deceived, that the characters were like a Mayan calendar or arranged in a column. In my view, Anthon was trying to separate himself from the matter, had no specific recollection of what Harris showed him, and is therefore less credible than Harris, to whom this had become a major factor in a life-changing decision.
Dean Jessee and others think the word "Caractors" written at the top is in the handwriting of Joseph Smith. Whether that means he personally wrote the characters or was just labeling the ones written by his wife, her brother, or Martin Harris, no one knows. Apparently Martin Harris kept a copy with him for years as evidence of the veracity of the Book of Mormon. Oliver Cowdery gave a copy to David Whitmer, which ultimately ended up at the Community of Christ.
Here are two views on all this history:
Stanley B. Kimball
Stout's criticism of Kimball
This cylinder was found at Tlatilco in the Valley of Mexico. Of the 28 characters identifiable on this cylinder, 26 have been shown to have a close relationship to Anthon characters.
Here is the Jones article making the comparison: http://www.shields-research.org/General/SEHA/SEHA_Newsletter_122-2.PDF
By contrast, the comparisons between the characters and the writing of the Mikmaq Indians of northeastern U.S. and Canada seem a closer fit. I suspect this comparison may account for whatever positive impression Mitchill and Anthon left with Harris. Of course, the last thing the Mesoamericanists want is a connection between the Anthon transcript and Native American Indian languages. This means we'll never see an article making this connection in the pages of the Interpreter, the Maxwell Institute, BYU Studies, etc., but we always have the Internet where people can make these comparisons and discuss them.
I haven't studied this enough to have a strong opinion--the only ancient languages I've studied are Latin and Greek--but to me, the Mesoamerican connection is very far-fetched, while the connection with the Mi'kmaq is undeniable. I know there are lots of reasons to question the antiquity of the Mi'kmaq characters,