The Empire had a population of trillions and was highly advanced technologically. The Tkon had the ability to move entire stars, using planets as outposts for defense. They employed guardians known as portals to defend these outposts and to act as gatekeepers for those seeking entry to the Empire.
Historical eras for this civilization were described as Ages measured by the Tkon using galactic motionary startime charts. The latter Ages of the Tkon Empire were Bastu, Cimi, Xora, and Makto. It was during the Age of Makto when the central star of their home system destabilized and went supernova, ending the Empire.
At least one of their guardians, Portal 63, survived and was encountered by the Federation starship USS Enterprise-D on a planet in the Delphi Ardu system in 2364. This occasion was the first instance of Federation contact with the extinct Tkon.
Beyond some information regarding planets in the Empire and an understanding of their script and some of their history, little was known about the Tkon by the Federation. Even a student of archaeology such as Jean-Luc Picard had not heard of the Tkon before 2364. (TNG: "The Last Outpost"; DIS: "Terra Firma, Part 2")
While on Dannus V in 3189, Carl showed Philippa Georgiou a copy of The Star Dispatch, which featured a headline on the front page that read "Supernova Threatens Tkon Empire". (DIS: "Terra Firma, Part 1", "Terra Firma, Part 2")
The writers of the episode "To the Death" mistakenly referred to the Iconians as the Tkon, causing the art department to design and produce sets based on graphics seen in "The Last Outpost". The error was discovered not long before shooting took place and corrected. (Star Trek: Deep Space Nine Companion (pp. 348-349))
In the Star Trek: The Next Generation - Q Continuum novels, Q reveals the history of the Tkon Empire's final days to Captain Picard while explaining Q's history with the malevolent entity 0. After Q allowed 0 and his allies access to this universe, he selected the Tkon Empire as a lesser species for 0 and his allies to 'test', regarding them as advanced enough to be interesting while (from the perspective of beings like 0) content to let their development plateau at this point without a suitable shove. As a result of 0's associates, the Tkon Empire was provoked into civil war between the central Tkon homeworld and its outer colonies over the costs of the "Great Endeavor", a plan to save their homeworld from its dying sun. Despite fifty years of conflict, the Tkon Empire not only survives but completes the Great Endeavor on schedule, intending to use an elaborate transporter array to teleport their sun completely out of their solar system and simultaneously replace it with a new, fresh one before their system could suffer from its loss. However, 0 triggers the collapse of their sun ahead of schedule in a fit of pique at the Tkon "winning" his "game", the elder Q noting to Picard that he at least adheres to his rules in his tests of Picard whereas 0 changed the rules the moment he was about to lose. The destruction of the Tkon Empire resulted in 0's banishment from the galaxy by the Q Continuum, as well as the creation of the galactic barrier to prevent him ever returning. Q notes to Picard that the transporter array the Tkon were going to use to move their new star into position still exists in the present, recommending that Picard salvage it once he returns to his own time before someone else does.