Zeke and Levi have been enemies for a long time. Levi hates Eren's half-brother Zeke because, in a battle of Shiganshima, he killed lot of handful soldiers along with Erwin of the Scout force.
Levi was driven by this to personally swear to avenge the fallen comrades, and one of his main objectives became to kill Zeke, the Beast Titan Shifter.
But Levi's desire for killing Zeke in the Attack on Titan finale isn't quite this simple; Zeke death serves a purpose far more important than taking revenge for the fallen.
Zeke essentially gave his life in this situation to make sure Eren couldn't use the Founding Titan ability and the Rumbling came to an end.
Even Levi was the one who had to kill Zeke, Zeke came out rather willingly because he understood what had to be done to protect the remaining Scouts and give them the opportunity to defeat Eren and stop his ambitious plan.
Even though killing Zeke was still one of Levi's goals, this may be argued that Levi's primary goal was to put an end to the Rumbling rather than just take vengeance.
Now, Levi can finally find closure in bearing the weight of the dead Scouts, allowing him to make peace with these terrible events—even though this still cannot make up for those he lost along the way.