First Drax

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

Youth
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) Upon becoming a renegade, Drax travelled the universe, doing repair and maintenance work in the fields of cybernetics, guidance systems and armaments. 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.

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 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. 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. Afterward, Drax stayed on Atrios to help rebuild that world. (TV: The Armageddon Factor)

The Doctor was later used as a fall guy in a con involving the third, fourth, fifth, sixth, seventh, eighth, ninth, tenth, eleventh, twelfth and thirteenth incarnation of Drax, who used a Blinovitch Limitation Effect limiter. The fifth Drax was using the identity "Galdron Cabot" (a criminal-turned-businessman), the eighth - "Fleur McCormick" (galactic police officer), the twelfth - "Charles Kirkland" (businessman and Cabot's rival), and the ninth - "Rosser" (Kirkland's butler). The fourth, sixth, seventh, tenth, eleventh and thirteenth incarnations were McCormick's "half a dozen best men". (AUDIO: The Trouble with Drax)

Alternative timeline
In an alternative 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.

During his 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. 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. The Doctor talked him round to joining him against the Shadow without much trouble. (TV: The Armageddon Factor)

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 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".