The letter from Barrow was sent to fellow gangster Raymond Hamilton, who was incarcerated at the Dallas (Texas) County Jail. According to RR Auction, the letter is not dated but is known to have arrived at the jail on April 30, 1934.
The auction house's website also shows a newspaper article from that time that explains how law enforcement confirmed the letter's authenticity. Some of Bonnie Parker's relatives identified her handwriting, and she was known to have written letters for Barrow.
The letter provides great detail about the lives of the gangsters. Hamilton had been a part of the Barrow Gang but had split with them a few weeks before the letter was written. In the letter, Barrow derides Hamilton's tendency to "live like a king" by sleeping in hotels and taking passenger trains. He also gets in a dig at Hamilton's girlfriend, referring to her as his "Prostitute Sweetheart."
You can read the entire text of the letter and see several pages of it at the auction company's website.