First Drax

Drax was a renegade Time Lord with a knack for tinkering and engineering. He knew the Doctor before either had left their home planet of Gallifrey.

Alternate timeline
In an alternate timeline Drax cooperated alongside the Master, the Monk, and the Rani to try to destroy the world using a DNA recombinator, turning the human race into a gestalt consciousness which could be used as a weapon to conquer the universe. (PROSE: The Quantum Archangel)

Personality
Drax showed great technological aptitude. However, he could not grasp temporal theory and failed the class. While Drax could not grasp temporal theory, he had worked out enough about the Blinovitch Limitation effect limiter to know its limitations and also to create an endless paradox from his third incarnation onwards to his thirteenth to possess the device and also commit numerous crimes with two or more of his future lives.

He also showed disrespect towards the Laws of Time. (AUDIO: The Trouble with Drax)

During his ten-year prison stay, Drax picked up Brixton's local accent and idioms, which remained part of his speech patterns after he left Earth and resumed his travels. This cockney accent would last for a number of regenerations, but by the twelfth, it would be gone. When Drax met K9 Mark II and marvelled at the "little tin dog", K9 noted Drax's silliness. Although Drax was not above criminal activity and weapons dealing for personal gain, he displayed a sense of what was morally right and bemoaned having to assist the Shadow in his schemes. He was quick to assure others that he didn't involve himself with armaments on a regular basis. The Doctor talked him round into joining him against the Shadow without much trouble. (TV: The Armageddon Factor, AUDIO: The Trouble with Drax)

Despite some differences, the incarnations of Drax seemed more amiable to one another compared to cases whenever the Doctors met, and despite the dangers of interacting with their own past and future they planned for events that happened in their future, such as having a female incarnation and later reminiscing about the gender switch being worth it. (AUDIO: The Trouble with Drax)

Behind the scenes

 * Although Drax's TARDIS is somewhere around in The Armageddon Factor, only components from it are seen on-screen. The interactive Make Your Own Adventure with Doctor Who book NOTVALID: Search for the Doctor included Drax as a character, and described his TARDIS as taking the shape of a convertible car, the registration plate of which read: "TARDIS".
 * According to NOTVALID: Dalek: Spoof Scenes, in a version of events where the Last Great Time War was fought against the Drashigs rather than the Daleks, the Ninth Doctor cited Drax as one of the people who'd perished in the War, alongside Damon, Spandrell and the Meddling Monk.