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, at which time he was known as Theta Sigma.

Youth
Like all Time Lords, Drax was taken from his family at the age of eight for the selection process in the Drylands. Staring into the Untempered Schism as part of a Time Lord initiation rite, Drax reacted by running away from what he saw in the Schism. (PROSE: A Brief History of Time Lords)

Drax attended the Prydonian Academy with the Doctor in the Class of 92. They took the tech course together. Though his practical work was satisfactory, he failed his exams, done in by temporal theory. (TV: The Armageddon Factor) Whilst he was at the academy, Drax became a member of the Deca and had his own self-made skimmer. (PROSE: Divided Loyalties)

As a renegade
Drax had already escaped from Gallifrey when the Doctor was expelled from the Time Lord Academy. (PROSE: Divided Loyalties) Unlike other renegades like his contemporaries, the Master and the Rani, Drax purchased his TARDIS instead of stealing one. (PROSE: Doctor Who and the Armageddon Factor)

Upon becoming a renegade, Drax travelled the universe, doing repair and maintenance work in the fields of cybernetics, guidance systems and armaments. He also committed cons. When Drax's TARDIS suffered a breakdown in its hyperbolics, Drax was stranded on Earth. He engaged in criminal activity to procure replacement parts and did time in Brixton prison for over ten years where he acquired a Cockney accent to help him survive inside, which would later become a trademark through several of his regenerations, though the Twelfth Drax would tell Romana II he had lost it a few regenerations ago, from that incarnation's perspective, during a con.

Later, the Black Guardian's agent, the Shadow, imprisoned Drax and forced him to build and maintain a giant computer called Mentalis on the deserted planet Zeos. Mentalis carried out a war against the neighbouring planet of Atrios.

450 years later in their personal timelines, Drax, then in his second incarnation, and the Doctor met again. While searching for the last segment of the Key to Time, the Fourth Doctor and Romana I arrived on Zeos. Drax, under the Shadow's orders, reluctantly led the Doctor into a trap. Drax claimed he hadn't expected the one looking for the Key to be the Doctor. Briefly, the two reminisced though at first, the Doctor needed a moment to remember him. Easily seeing through Drax's pretence, the Doctor talked him into joining forces against the Shadow. With Drax's assistance, the Doctor recovered the final segment of the Key to Time. Afterwards, Drax stayed on Atrios to help rebuild that world. (TV: The Armageddon Factor) During that time Drax had hacked the directional unit of the Doctor's TARDIS.

Later, after he had regenerated for the second time, the Third Drax gained the map to Altrazar. The Fourth Doctor was later used as a fall guy in a con involving eleven of Drax's incarnations, from the third to thirteenth incarnations, who used a Blinovitch Limitation Effect limiter. The ninth Drax was using the identity "Galdron Cabot" (a criminal-turned-businessman), the eighth — "Fleur McCormick" (galactic police officer, the only female incarnation of Drax), the twelfth — "Charles Kirkland" (businessman and Cabot's rival), and the fifth — "Rosser" (Kirkland's butler). The fourth, sixth, seventh, tenth, eleventh and thirteenth incarnations were Inspector McCormick's "half a dozen best men". The Doctor was conned into stealing the Blinovitch Limitation Effect limiter, which was what allowed so many Draxes to successfully cooperate on this scheme and others in the first place. By the time the Doctor and Romana realised that Drax must have known he would have been arrested over different incarnations, they realised the truth about McCormick and her men, and by the time they returned all of Drax's incarnations had escaped in their TARDIS. (AUDIO: The Trouble with Drax)

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.