Even though Slade Wilson, Deathstroke the Terminator, and Green Arrow have been at each other's throats for years, it might not be readily apparent why these two are such fierce rivals. Deathstroke originally appeared as a villain in New Teen Titans #2, vowing vengeance against them because he blamed the teenagers for the death of his son, Grant. Grant had undergone genetic modifications that ultimately killed him, which led to Deathstroke taking his frustration out on the teens.
However, while Deathstroke's rivalry with Robin, Starfire and the rest dates back to his origins, Deathstroke's rivalry with Oliver Queen is much more recent. The rivalry sparked during the events of Brad Meltzer and Rags Morales' Identity Crisis, when Doctor Light hired Deathstroke to protect him from the Justice League. In an event that had so little to do with Deathstroke personally, Wilson met the man who would prove to be one of his greatest enemies.
Green Arrow Managed to Best Deathstroke
In a notoriously one-sided battle, Deathstroke defended Doctor Light from the entire Justice League, easily dismantling them one by one. He runs the Flash through on his sword, overpowers Green Lantern's will with his own, and it seems as though he has a tactical answer for anything the League is capable of throwing his way. However, this changes when Green Arrow leaps on his back and jabs a broken arrow right through Deathstroke's hollowed-out eye socket.
This attack manages to anger Deathstroke so greatly that, at the end of the story, Slade Wilson returns the arrow to Green Arrow with a single note: "This is yours. We're not done." This thinly-veiled declaration of vengeance showed that Deathstroke took the attack personally -- or, at the very least, saw Green Arrow's skill worthy enough to merit a rematch.
Deathstroke later learned that Oliver Queen is the Green Arrow, and waited for the right time to strike. The time came, as it turned out, after the post-Infinite Crisis time jump. Deathstroke is hired by several businessmen to slaughter Star City's mayor, who happens to be Queen. Recognizing Green Arrow as the man who humiliated him, Deathstroke pursues him. However, Oliver Queen, using both his authority as the mayor of Star City and his abilities as Green Arrows, manages to capture Deathstroke.
However, this ultimately proves to be a massive feint on Deathstroke's part. He allows himself to be captured in order to gain information on Green Arrow from the jailed villains.
Deathstroke's Vendetta Against Green Arrow Gets Worse
Deathstroke's vendetta against Green Arrow would only snowball over time, with there being indications that he wanted to rip Green Arrow and his life apart, now knowing everything he needed to know about Oliver Queen to dismantle his entire life.
Deathstroke attempted to convince the hero Geo-Force to become a spy within the Justice League, threatening an environmental crisis with powers he might or might not have had. Geo-Force learned, however, that Deathstroke planned to use him as a pawn not to conquer the world or get a lot of money, but rather to gain information necessary for Slade to crash Oliver Queen's wedding to Dinah Lance, aka the Black Canary.
Ultimately, Deathstroke did bring a supervillain team down on the superhero wedding, though because the wedding was full of superheroes, the villains were quickly subdued and captured, with Oliver and Dinah tying the knot in the process. Deathstroke managed to get away, failing in his mission but confirming that his grudge against Green Arrow had reached such levels of absurdity and pettiness that he felt content to just ruin Oliver's day.