As detailed in the films, Drax and Thanos have a history rooted in the murder of Drax's family. However, in the comics, this is even more overt. Originally, Drax wasn't even an alien.
Originally, Drax was a human named Arthur Douglas, whose family just happened to have the misfortune of spotting Thanos' ship during his first scouting trip to Earth. To prevent any witnesses, Thanos killed all of them, including Arthur Douglas.
Around the same time, Kronos (father of the Gods) was engineering a way to stop Thanos for Mentor (Thanos' father). This resulted in Kronos programming a body and putting Douglas' soul inside of it, creating Drax the Destroyer, and giving him one purpose in life: to destroy Thanos.
Drax finally fulfilled this purpose in Annihilation #4, which saw Thanos teaming up with fellow-baddie, Annihilus. After slicing his way through Thanos' army, he punches through Thanos' back and rips out his still-beating heart.