Drex had a reputation for belligerence, and frequently harassed station personnel and civilians while he was at Deep Space 9. He and two accomplices were responsible for accosting Elim Garak in his shop. Garak declined to press charges, noting that he got off several remarks that caused "serious damage to their egos".
Suspicious of their motives, Captain Benjamin Sisko ordered Lieutenant Commander Worf to investigate. Seeking to gain Martok's attention, Worf assaulted Drex in Quark's and took his d'k tahg. (DS9: "The Way of the Warrior")
Drex was played by Obi Ndefo.
The name "Drex" was coined by Ira Steven Behr after reading the nickname on production staffer Doug Drexler's crew jacket, and which was a source of pride for Drexler, "I got a Klingon named after me! There was a Klingon named Drex. Michael Dorn always loved my last name for a Klingon; on TNG, he would yell across the stage, 'Drexler, Drexler, Drexler.' He'd say, 'It's a great Klingon name.' And I think I told Ira that. Also, I had Drex on my jacket. Ira always used to see that." (Star Trek: The Magazine Volume 1, Issue 22, p. 88)
Drex was mentioned in the pronunciation guide of the script of the sixth season episode "Sons and Daughters". It is possible that Drex appeared or was mentioned in an early version of the script. Ultimately, much like Zek's son Krax, Drex was never seen or directly mentioned again after his first appearance. Ronald D. Moore commented that the writers discussed bringing back Drex in the final season. 
In "You Are Cordially Invited," Martok describes his wife Sirella as the mother of his children, so Drex must have had at least one sibling.
Drex appeared as first officer of the IKS Gorkon in the novel Diplomatic Implausibility, the first book of what later become a series of novels about that ship by Keith R.A. DeCandido. The captain of the Gorkon, Klag, was of the belief that Drex had inherited none of his father's honor, and was a detriment to the ship and its crew; he was replaced by Tereth shortly before The Brave and the Bold, Book Two.
In the story of Star Trek Online, Drex has become head of the House of Martok after Martok's apparent death and Worf serves as gin'tak for the House, advising Drex on how to successfully lead the House. Drex's son, M'ven, is the heir to the House, until he is murdered by the House of Torg and the Tal Shiar at the bat'leth competition on Forcas III. A Tal Shiar report from much later in the game heavily implies that Drex is intimate with, and thus being manipulated by, one of the matriarchs of the Orion Syndicate, but he does not appear at any point in the game.